TECHNICAL APPENDIX - NOT FOR PUBLICATION

Immigration, Remittances and Business Cycles Technical Appendix and Additional Results Federico Mandelman and Andrei Zlate1

This appendix is divided in four sections: Section A. Baseline Model This section presents supplemental material and results for the baseline model estimated in the paper. It includes: (1) Steady state: derivation of the baseline model’s steady state. (2) Bayesian estimation: description of the data sources, the choice of prior distributions, the estimation methodology, the empirical evaluation of the model performance, and the Kalman smoothing procedure. (3) Model …t: we compare the model’s Kalman …ltered one-sided predicted values with the data; we also compare the vector autocovariances from the simulated model series with those from the actual data. (4) Forecast error variance decomposition: examine the relative contribution of shocks to ‡uctuations in the model variables. (5) Sensitivity analysis: the benchmark model is estimated with data that is detrended using a linear trend (rather than cubic trend as in the paper). (6) Additional estimation results: Plots of the prior and posterior densities of the parameters of the benchmark model, the median impulse responses to each of the model shocks, including the 10 and 90 percent posterior intervals. (7) Markov Chain Monte Carlo (MCMC) univariate and multivariate convergence diagnostics. Section B. Remittances (1) This section describes the microfoundations of the altruistic mechanism for remittances. It also presents the sensitivity analysis to alternative scenarios: (2) The relative allocation of consumption within the foreign household (across resident and emigrant members) departs from the assumption of equal consumption per worker in the baseline model, thus resembling the relative bargaining power of residents vs. emigrants; (3) The share of unskilled household in foreign capital ranges from low to high; less capital holdings for the unskilled implies less resources available to invest in migration, which in turn bolsters the insurance mechanism of the remittances; (4) Remittances depend on the Home-Foreign output di¤erential, mimicking Acosta et al. (2009). Section C. Alternative Model with Financial Integration for Unskilled This section: (1) Describes the alternative in which both skilled and unskilled households trade bonds internationally. (2) Examines the implications of …nancial integration for the unskilled for the cyclical 1

Beyond the usual disclaimer, we must note that any views in this paper are solely the responsibility of the authors and should not be interpreted as re‡ecting the views of the Federal Reserve Bank of Atlanta, the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

1

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behavior of migration and remittances. (3) Uses the Bayes factor to show that the baseline model in which the unskilled are in …nancial autarky …ts the data better than the alternative model with full …nancial integration presented in this appendix. Section D. Alternative Model with Social Planner This section: (1) Presents and alternative model with a social planner in each economy and …nancial autarky, like in the …rst version of the paper. However, the model includes skilled and unskilled households in each economy, with capital-skill complementarity in production. (2) Presents the estimation results, and compares them with those in the previous manuscript.

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Immigration, Remittances and Business Cycles SECTION A: BASELINE MODEL 1

Steady State for Baseline Model

In this section, variables without time subscripts represent steady-state values. We calibrate the steadystate values of labor ls = Ls =s = 0:5, lu = Lu =(1

s) = 0:5, ls = Ls =s = 0:5 and lu = Lu =(1

and compute the corresponding weights on the disutility from labor

s,

u,

and

s

u.

s ) = 0:5,

From the Euler

equations for capital, we obtain the gross rental rate of capital (which includes depreciation) as: rs = rs = ru = ru = +(1 rb = (1

) = : From the Euler equations and the market clearing conditions for bonds, we obtain Bh = 0 for the home bonds, and rb = (1

) = and Bh =

) = = rb and Bf =

Bf = 0 for

the foreign bonds. From the law of movement of the stock of capital, we get is = ks , iu = ku , is = ks and iu = ku . From the law of movement of the stock of immigrant labor, we get le = [ l =(1

l )] li .

We solve the steady state numerically using a system of 24 unknowns (Yeh , Yef , K, K , Yh , Yf , ph , pf ,

Q, Li , Ci , Y , Y , ws , wu , ws , wu , Ks , Ks , Ku , Ku , Cs , Cs and Cu ) and 24 non-linear equations (1-24 below). Output in Home and Foreign are: Yeh

Yef

1

= 1

=

1

1

(Lu + Li ) 1

(Lu

+ (1 1

Li )

)

1

h

+ (1

)

1

1

K 1

1

1

1

+ (1

) ( Ls ) 1

K

+ (1

)

i

1

(

1

)

(1) (

1

1

; (

1) 1)

( Ls )

:

(2)

Aggregate capital in Home and Foreign are: K K

1

k k

=(

k)

=(

k)

1

k k

1 k

(Ks;t )

1 k

1

k k

k)

+ (1

k)

1

k

Ks;t

+ (1

k

1 k

(Ku;t )

1 k

Ku;t

1

k k

;

(3)

:

(4)

1

k k

The gross rental rates of capital owned by the skilled households in Home and Foreign are:

rs = ph (1 rs = pf (1

)

1

1

)

1

1

Yeh

1

h Yef

1

1

1

K

+ (1 1

1

1

) ( Ls ) 1

K

+ (1

)

3

1

i(

1)

K

1 k

Ks 1

( Ls )

kK

1

(

1)

K

;

(5) 1

kK

Ks

1 k

:

(6)

TECHNICAL APPENDIX - NOT FOR PUBLICATION

The gross rental rates of capital owned by the unskilled households in Home and Foreign are:

ru = ph (1 ru = pf (1

)

1

1

)

h

1

Yeh

1

1

1

Yef

1

K

+ (1

1

1

1

1

) ( Ls ) 1

K

+ (1

1

)

i(

1)

(1

1

K

1 k

k )K

Ku (

1

1)

( Ls )

1

K

; (1

(7) 1 k

k )K

Ku

: (8)

The skilled wages in Home and Foreign are: ws = ph (1 ws = pf (1

)

1

1

1

1

) Y~h

) (1 (1

)

1

h Y~f

1

1

K

1

1

1

1

+ (1

) ( Ls ) 1

K

+ (1

)

i(

1

1)

1

1

( )

Ls ; 1

(

1)

( Ls )

(9) 1

1

Ls

:

(10)

The unskilled wages in Home and Foreign are: Y~h Lu + Li !1

wu = wi;t = ph

Y~f Lu Li

wu = pf

!1

;

(11)

:

(12)

The budget constraint of the representative unskilled household in Home is: wu lu + ru ku = cu + iu ; where consumption of the unskilled household is cu = [Y

(13)

Ks

Ku

Ci

Cs ] = (1

s) :

The budget constraint of the unskilled household in Foreign is: wu lu + wi Q

1

li

wu li + ru ku = cu + fe wi Q

1

le + iu :

(14)

The demand ratios for the two intermediate goods in each country are: Yeh

Yef

Yh Yf Yf Yh

= =

! 1

! !

1

!

4

ph pf Q pf Q ph

;

(15) :

(16)

TECHNICAL APPENDIX - NOT FOR PUBLICATION

The price indexes for the composite good of each country are: 1 = ! (ph )1

!)(pf Q)1

+ (1

1 = ! (pf )1

;

(17)

! ) (ph =Q)1

+ (1

:

(18)

The composite goods in Home and Foreign are: 1

Y

1

1

=! 1

(Y )

= (! )

Yeh 1

Yh

1

1

+ (1

!) (Yf )

1

Yef

Yf

+ (1

(19) 1

1

! )

(Yh )

(20)

Next, we need an expression for the steady-state ratio between the immigrant wage in Home and the unskilled wage in Foreign. From the net present value of the gains from emigration, fe Q 1

(1 l ) (1 l )

wi Q

1

1w i

=

wu , we obtain: wi = 1 wu Q

Note that when fe = 0, it follows that

1

(1 (1

l) l)

1

fe

:

= 1. In other words, when the sunk emigration cost is zero, labor

migration will take place until, in equilibrium, the immigrant wage in Home is equal to the unskilled wage in Foreign. Next, we use the expressions for wi and wu from (10) and (12) into the previous equation to obtain: ph |

Yeh Lu + Li {z wi

!1

= pf

}

|

Y~f Lu Li {z

!

wu

1

Q:

(21)

}

Given that Bh = Bf = 0, the expression for the current account becomes:

ph Yh

pf QYf = Li ph |

The resource constraint of the foreign economy is: Y = Cs + Cu

Ci Q

1

Yeh Lu + Li {z wi

!1

Ci :

(22)

}

+ Is + Iu + fe wi;t Q

1

Le

(23)

Finally, the assumption that members of the foreign unskilled household (residents and migrants) enjoy

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equal consumption per capita implies: Cu Ci Q Lu Li

1

=

Ci Q Li

1

:

(24)

Next we use the steady-state solutions above to obtain solutions for the remaining variables: Le = [ l =(1

l )] Li ;

Yf = Yef

and

2 2.1

u

=

v = fe wi Q

1;

d = v1

(1 l ) (1 l ) ;

Cu = Y

Ks

Yf . Finally, the weights on the disutility from labor are wu cu (lu )

Ku s

=

Cs ws cs (ls )

,

Ci ; Yh = Yeh u

=

wu cu (lu )

,

Yh ; and s

=

ws cs (ls )

.

The Bayesian Estimation Data Sources

Macroeconomic data on real output, consumption and investment is provided by the Bureau of Economic Analysis (for the U.S.) and INEGI (for Mexico) through Haver Analytics. We use the multilateral (global) trade balance for Mexico, which is available for the entire sample period, since the bilateral trade balance with the U.S. is only available starting in 1990:Q1. (However, the correlation of the detrended bilateral and multilateral series is 0.97 for the overlapping period). Data on apprehensions at the U.S.-Mexico border and the number of hours spent by the U.S. border patrol policing the border is provided by the U.S. Immigration and Naturalization Service, and made available on Gordon Hanson’s website (“border linewatch apprehensions” and “border linewatch enforcement hours”). Data on workers’ remittances in nominal U.S. dollars is provided by the Bank of Mexico through Haver Analytics. We convert remittances in real dollars using the U.S. CPI index, and then in real Mexican pesos using the U.S.-Mexico bilateral real exchange (obtained from the IMF’s International Financial Statistics). We de…ne the skill premium as the ratio between the average hourly wage of workers with 12 or more years of schooling (weighted by shares in labor force) and the average hourly wage of workers with less than 12 years of schooling. To calculate the U.S. skill premium, we use the U.S. Census Bureau’s Current Population Survey (CPS), the Merged Outgoing Rotation Group (MORG). Data on total hours worked last week, weekly earnings before deductions, the appropriate earnings weights, and educational attainment have been collected for the years 1980 to 2004.2 For the Mexican skill premium, we compute the skilled and unskilled wages as weighted averages of the hourly labor income of male Mexican residents with more/less than 12 years of education, using the data from Hanson (2006). Data on hourly wages for Mexico’s maquiladora sector (in nominal 2 The hourly wage data are available for four education groups: (a) no high school degree; (b) completed high school; (c) some college or associate’s degree; and (d) bachelor’s degree or higher. We compute the skilled wage as the weighted average of hourly labor incomes across education groups (b), (c) and (d). The unskilled wage corresponds to group (a).

6

TECHNICAL APPENDIX - NOT FOR PUBLICATION

pesos) is provided by INEGI; we compute the hourly wage in nominal pesos using the monthly series on total remunerations, total number of workers, and hours per worker. In order to compute the ratio between the U.S. unskilled-Mexican maquiladora wage, we convert the maquiladora wage in U.S. dollars using the nominal exchange rate. We seasonally adjust the data with the X-12 ARIMA method of the U.S. Census Bureau.

2.2

Prior Distributions

This section adds to the discussion of the choice of prior distributions for our estimated parameters, which are presented in Table 1 in the main paper. As discussed in the paper, we base the assumption that and

>

>

on the …ndings of Krusell et al. (2000) that skilled labor and capital are relative complements.

Krusell et al. (2000) document a high complementarity between skilled labor (i.e. college graduates) and capital ( = 0:67) relative that between unskilled labor and capital (1:67). Since our pool of skilled workers is much larger (i.e. high school degree or more), we center the priors for

= 0:85 and

above the value in Krusell et al. (2000), but still below the priors assigned to

=

= 0:73 slightly

= 0:95: Based on the

capital-skill complementary assumption, we choose rather tight priors for these parameters. We do not have much prior information about the magnitude of shocks. Therefore, the variances of all shocks are harmonized as in Smets and Wouters (2007), and assumed to follow an Inverse Gamma distribution that delivers a relatively large domain. The autoregressive parameters in the shocks are assumed to follow a Beta distribution that covers the range between 0 and 1. For these, we select rather strict standard deviations and thus have tight prior distributions in order to obtain a clear separation between persistent and non-persistent shocks, and also to generate volatilities for the endogenous variables that are broadly in line with the data (see Smets and Wouters, 2003, for details). For the remaining parameters we consider Beta or Gamma distributions, which are restricted to the positive support. We set a relatively loose prior for the elasticity of substitution between the home and foreign goods

centered at 1.5, the value in Backus et al. (1994). We set the prior mean of

at 1, which

delivers a Frisch elasticity of labor supply that is in between microeconomic estimates and the relative larger values usually observed in the macro literature. We also set for the skilled),

k

=

k

k

= 0:96 and

k

= 0:9 (capital shares

= 1:5 (substitution in the CES capital composite), =s = 0:45 and

=s = 0:45

(adjustment cost for per capita bond holdings). In the technical appendix, we discuss the e¤ect of k

on the results.

7

k

and

TECHNICAL APPENDIX - NOT FOR PUBLICATION

2.3

Estimation Methodology

In this section we brie‡y explain the estimation approach used in this paper. A more detailed description of the method can be found in Lubik and Schorfheide (2005), Justiniano and Preston (2010), and Schorfheide (2000) among others. Let’s de…ne

as the parameter space of the DSGE model, and z T = fzt gTt=1 as

the data series used in the estimation. From their joint probability distribution P (z T ; ), we can derive a relationship between the marginal P ( ) and the conditional distribution P (z T j ); which is known as the Bayes theorem: P ( jz T ) / P (z T j )P ( ): The method updates the a priori distribution using the likelihood contained in the data to obtain the conditional posterior distribution of the structural parameters. The resulting posterior density P ( jz T ) is used to draw statistical inference on the parameter space

.

Combining the state-form representation implied by the solution for the linear rational expectation model and the Kalman …lter, we can compute the likelihood function. The likelihood and the prior permit a computation of the posterior that can be used as the starting value of the random walk version of the Metropolis-Hastings (MH) algorithm, which is a Monte Carlo method used to generate draws from the posterior distribution of the parameters. In this case, the results reported are based on 500,000 draws following this algorithm. We choose a normal jump distribution with covariance matrix equal to the Hessian of the posterior density evaluated at the maximum. The scale factor is chosen in order to deliver an acceptance rate between 30 and 45 percent depending on the run of the algorithm. Measures of uncertainty follow from the percentiles of the draws.

2.4

Empirical Performance

We de…ne the marginal likelihood of a model A as follows: MA =

R

P ( jA)P (z T j ; A)d , where P ( jA)

is the prior density for model A, and P (z T j ; A) is the likelihood function of the observable data, conditional on the parameter space

and the model A. The Bayes factor between two models A and B is the de…ned

as: FAB = MA =MB .3 The marginal likelihood of a model (or the Bayes factor) is directly related to the T +m R predicted density of the model given by: p^TT +m P ( jz T ; A) P (zt jz T ; ; A)d : Where p^T0 = MT : +1 = t=T +1

Therefore the marginal likelihood of a model also re‡ects its prediction performance.

2.5

Smoothing

The DSGE model can be written in a state-space representation as in which

t

t+1

=F

t

+ vt+1 and zt = H 0

t

+ wt ,

is the vector of unobserved variables at date t, and zt is the vector of observables; shocks vt and

wt are uncorrelated, normally distributed, white noise vectors. The …rst expression is the state equation 3

Notice that ln(FAB ) = log(MA =MB ) = log(MA ) of the log marginal likelihood of each speci…cation.

log(MB ):That is the Bayes Factor may be interpreted as the di¤erence

8

TECHNICAL APPENDIX - NOT FOR PUBLICATION

and the second the is the observed equation. Smoothing involves the estimation of

T

= f t gTt=1 conditional on the full data set z T used in the

estimation. The smoothed estimates are denoted

tjT

= E( t jz T ) and, as shown in Bauer et al. (2003),

can be written as: tjT

=

1 + Ptjt F 0 Pt+1jt 0 t+1jt )

t+1jT

t+1jt

;

in which Pt+1jt = E(

t+1

the projection of

on z t and a constant, projection which is denoted as

t+1

t+1jt )( t+1

tjt

the Kalman Filter to calculate f t gTt=1 ; estimates

3

tjT

T t=1

(25)

is the mean squared forecasting error associated with

T 1 t+1jt t=0 ;

T Ptjt t=1

and

t+1jt

T 1 Pt+1jt t=0 ;

= E(

t+1 jz

t ).

Using

the sequence of smooth

is determined from equation (25).

Model Fit

3.1

Kalman-Filtered One-Sided Estimates

Fig. A1 reports the benchmark model’s Kalman …ltered one-sided estimates computed at the posterior (dashed line) together with the data (solid line). The model …t appears to be satisfactory with one exception: at the very beginning of the sample period, the …tting for US investment and US consumption is not adequate. This may be the by-product of an identi…cation issue. Namely, the discount factor rate and investment-speci…c shocks imply similar aggregate dynamics for the Home economy, as they hit the Home aggregate demand in a similar fashion. Nonetheless, as more past observations enter in the …lter, the model predictions gain accuracy.

3.2

Autocovariance Functions

To further assess the model adequacy, we compare the vector autocovariance functions in the model and in the data, as in Adolfson et al. (2007). The function depicts the covariance of each observable variable against itself (measured at lags h = 0; 1; :::5) and other variables. These functions are computed by estimating an unrestricted VAR model with both the U.S.-Mexico data and arti…cial data sets of the same time length generated through model simulations with parameter draws from the posterior. We include output for both economies, border enforcement and apprehensions/migration ‡ows.4 Fig. A2 displays the median vector autocovariance function from the DSGE speci…cation (thin line), along with the 2.5 and 97.5 percentiles for the mentioned subset of variables. The posterior intervals for the vector autocovariance 4

We draw 3,000 parameter combinations from the posterior distribution and simulate 3,000 arti…cial data sets (of the same length as the ones in the data) to estimate vector autocovariance functions using the same VAR speci…cation applied on the actual U.S./Mexico data.

9

TECHNICAL APPENDIX - NOT FOR PUBLICATION

are wide. In this case, this range re‡ects both parameter and sample uncertainty, which in the latter case is the result of using relatively few observations in the computations. Nonetheless, in general, the data covariances (thick lines) fall within the error bands, indicating that the model is somewhat able to replicate the cross-variances in the data. Overall, the model …t is satisfactory, particularly when taking into consideration that neither migration ‡ows nor remittances are part of the data that we use in the Bayesian estimation.

4

Forecast Error Variance Decomposition

Fig. A3 displays the forecast error variance decomposition of the seven observables used in the estimation, plus remittances and migration ‡ows at various horizons (Q1, Q4, Q16, Q40), based on the posterior benchmark estimation. Technology shocks aggregate the e¤ects of both neutral and investment-speci…c shocks. As discussed in Justiniano et al. (2009), the investment shock a¤ects the marginal e¢ ciency of investment, and it can be linked to more fundamental disturbances to the functioning of the …nancial system that negatively a¤ect productivity but are not explicitly included in the model. Shocks to border enforcement, which are assumed to be exogenous, and demand (discount rate) shocks complete the list. In the short run (within a year), demand shocks play a primary role in driving the dynamics of the real macroeconomic variables in the U.S., which is in line with evidence in Smets and Wouters (2007). While the demand shocks are relatively less important to explain output in Mexico at short horizons, their impact on consumption is nonetheless sizable. In the medium to long-run, technology shocks are dominant drivers of output, consumption and investment in both countries. At all horizons, Mexican technology and border enforcement shocks are the most important drivers of labor migration ‡ows. In the short run, technology innovations in Mexico dominate (explain more than 70% of the forecast error variance at Q1), whereas the border enforcement shocks become relatively more important in the medium to long-run. Technology shocks in the U.S. play a negligible role in the migration dynamics, which is consistent with the lock-in e¤ect that makes the stock of immigrant labor non-reactive to macroeconomic developments in the destination economy. At short horizons, technology shocks and, to some extent, demand shocks explain practically all the variability of remittances. Remittances are reactive to shocks originating on both sides of the border, a result which highlights its insurance role. Instead, in the long-run, the forecast error variance of remittances is signi…cantly in‡uenced by shocks to border enforcement, indicating that remittances are linked to the labor migration developments at longer horizons.

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TECHNICAL APPENDIX - NOT FOR PUBLICATION

5

Sensitivity Analysis

For robustness, in Table A1 we report the estimation results obtained for the baseline model, but using linearly detrended data. In contrast, the paper reports the estimation results with the data in deviations from a cubic trend.

6

Additional Results

In Figure A4, we show the prior (grey line), posterior density (black line) and mode from the numerical optimization of the posterior kernel (dashed line) for the benchmark model. Figure A5 reports impulse responses to all shocks (neutral technology, demand and investment in either Home or Foreign, as well as border enforcement). We depict the median response (solid lines) to a one standard deviation of the shocks, along with the 10 and 90 percent posterior intervals (dashed lines).

7

Convergence Diagnostics

We monitor the convergence of iterative simulations with the methods described in Brooks and Gelman (1998). General Univariate Diagnostics The empirical 80 percent interval for any given parameter, %, is taken from each individual chain …rst. The interval is described by the 10 and 90 percent of the n simulated draws. Then, m within-sequence interval length estimates are constructed. Next, a set of mn observations, generated from all chains, is also used to calculate the 80% interval, and a total-sequence interval length ^ interval = estimate is obtained, so that R

length of total-sequence interval mean length of the within-sequence interval

can be evaluated: Convergence

^ ! 1): is approached when the numerator and denominator coincide (i.e. R It is also possible to compute non interval-based alternatives, which we report for robustness. The numerator and denominator in the expression above is replaced by an empirical estimate of the central sth order moments calculated from all sequences together, and the mean sth order moment is calculated from ^s = each individual sequence, so as to de…ne for every s: R

Pm Pn 1 t=1 j%jt mn 1 Pj=1 m Pn 1 j=1 t=1 j%jt m(n 1)

%::js %j :js

: In Figure A6, we plot

^ interval ; R ^2; R ^ 3 for each of the parameters estimated: the numerator and denominator from measures of R The scale used for drawing the initial value of the MH chain is twice that of the jumping distribution in the MH algorithm. As it is observed, convergence is achieved before 100,000 iterations (we use …ve parallel chains) for most parameters. Instead, signi…cantly more draws are needed to achieve convergence for and 5

.5

The convergence diagnostic charts are available upon request (including those for the parameters characterizing the shock

11

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Multivariate extensions In this case, we rede…ne % as a a vector parameter based upon observations (i) %jt

denoting the ith element of the parameter vector in chain j at time t: The direct analogue of univariate

approach in higher dimensions is to estimate the posterior variance-covariance matrix as: V^ = nn 1 W + Pn P P 1 %j :)(%jt %j :)0 and B=n = m1 1 m (1 + m %:: )(%j: %:: )0 : )B=n; where W = m(n1 1) m t=1 (%jt j=1 j=1 (%j: It is possible to summarize the distance between V^ and W with a scalar measure that should approach

1 (from above) as convergence is achieved, given suitably overdispersed starting points. We can monitor both V^ and W; determining convergence when any rotationally invariant distance measure between the two matrices indicates that they are su¢ ciently close. In Figure A7, we report measures of this aggregate.6 Convergence is achieved before 100,000 iterations.

References [1] Adolfson, M., Laseen, S., Lindé, J. and Villani, M. (2007). "Bayesian Estimation of an Open Economy DSGE model with Incomplete Pass-Through," Journal of International Economics, 72(2): 481–511. [2] Backus, D., Kehoe, P. and Kydland, F. (1994). "Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?" American Economic Review, 84(1): 84–103. [3] Bauer, A. , Haltom, N., and Rubio-Ramírez, J. (2003). "Using the Kalman Filter to Smooth the Shocks of a Dynamic Stochastic General Equilibrium Model," Federal Reserve Bank of Atlanta, Working Paper 2003-32. [4] Brooks, S., Gelman, A. (1998). "General Methods for Monitoring Convergence of Iterative Simulations," Journal of Computational and Graphical Statistics 7(4), 434–455. [5] INEGI (El Instituto Nacional de Estadística y Geografía, Mexico, 2008). Banco de Información Económica, http://dgcnesyp.inegi.org.mx/cgi-win/bdieintsi.exe [6] Justiniano, A. and Preston, B. (2010). "Monetary Policy and Uncertainty in an Empirical Small Open Economy Model," Journal of Applied Econometrics, 25(1). [7] Justiniano, A., Primiceri, G., Tambalotti A. (2009). "Investment Shocks and the Relative Price of Investment," CEPR Discussion Paper No. 7597. processes). 6 Note that, for instance, the interval-based diagnostic in the univariate case becomes now a comparison of volumes of total and within-chain convex hulls. Brooks and Gelman (1998) propose to calculate for each chain the volume within 80%, say, of the points in the sample and compare the mean of these with the volume from 80% percent of the observations from all samples together.

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[8] Krusell, P.; Ohanian, L., Rios-Rull, J. and Violante, G. (2000). "Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis," Econometrica, 68(5): 1029–1054. [9] Lubik, T. and Schorfheide, F. (2005). "A Bayesian Look at New Open Economy Macroeconomics," NBER Macroeconomics Annual, 313–366. [10] Schorfheide, F. (2000). "Loss Function Based Evaluation of DSGE Models," Journal of Applied Econometrics 15(6), 645–670. [11] Smets, F. and Wouters, R. (2007). "Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach," American Economic Review, 97(3): 586-606. [12] Smets, F. and Wouters, R. (2003). "An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area," Journal of the European Economic Association, 1(5): 1123–1175. [13] U.S. Census Bureau (2007). Current Population Survey (CPS), Annual Social and Economic Supplement.

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Table A1: Summary statistics for baseline model estimated with linearly detrended data Prior distribution Description

Name

Posterior distribution

Density

Mean

Std Dev

Sd (Hess)

Mode

Mean

5%

95%

Gamma

0.06

0.01

0.0080

0.0621

0.0556

0.0407

0.0669

E la st. o f su b st. (K , u n sk ille d ).

Beta

0.95

0.015

0.0147

0.9507

0.9571

0.9371

0.9802

E la st. o f su b st. (K , sk ille d )

Beta

0.85

0.015

0.0099

0.9061

0.9042

0.8880

0.9198

P ro d u c t. o f n a tive sk ille d

Gamma

7

0.75

0.1650

7.5531

7.5430

7.3150

7.8299

S h a re o f u n sk ille d in o u tp u t (F )

Gamma

0.40

0.01

0.0042

0.3962

0.3949

0.3875

0.4019

E la st. o f su b st. K , u n sk ille d , (F )

Beta

0.95

0.015

0.0058

0.9620

0.9568

0.9467

0.9654

E la st. o f su b st. K , sk ille d (F )

Beta

0.73

0.015

0.0061

0.7747

0.7844

0.7755

0.7951

P ro d u c t. o f n a tive sk ille d (F )

Gamma

5.2

0.75

0.1482

5.4679

5.0550

4.8048

5.2622

Gamma

2.8

0.30

0.0000

5.1494

5.1494

5.1494

5.1494

Inve rse e la st. o f la b o r su p p ly

Gamma

1

0.2

0.0916

1.5121

1.5572

1.4005

1.7152

E la st. o f su b stitu tio n , g o o d s

Gamma

1.5

0.3

0.2043

1.8436

1.7969

1.4955

2.1387

Gamma

0.99

0.1

0.0692

0.9774

1.0548

0.9210

1.1604

S h a re o f u n sk ille d in o u tp u t

S u n k e m ig ra tio n c o st

E la st. o f re m itta n c e s to w a g e s

fe

'

N e u tra l te ch . sh o ck (H )

a

Beta

0.75

0.1

0.0265

0.9373

0.9282

0.8895

0.9755

N e u tra l te ch . sh o ck (F )

a

Beta

0.75

0.1

0.0263

0.9479

0.9380

0.8978

0.9818

D isc o u nt fa c to r sh o ck (H )

b

Beta

0.5

0.05

0.0198

0.7690

0.7618

0.7336

0.7902

Inve st. te ch . sh o ck (H )

i

Beta

0.5

0.05

0.0000

0.7902

0.7902

0.7902

0.7902

D isc o u nt fa c to r sh o ck (F )

b

Beta

0.5

0.05

0.0150

0.7894

0.7710

0.7492

0.7902

Inve st. te ch . sh o ck (F )

i

Beta

0.5

0.05

0.0299

0.7152

0.7276

0.6800

0.7763

B o rd e r e n fo rc e m e nt sh o ck

fe

Beta

0.75

0.1

0.0046

0.9927

0.9898

0.9831

0.9972

N e u tra l te ch . sh o ck (H )

a

Inv gamma

0.01

2*

0.0005

0.0065

0.0066

0.0058

0.0075

N e u tra l te ch . sh o ck (F )

a

Inv gamma

0.01

2*

0.0015

0.0171

0.0172

0.0149

0.0196

D isc o u nt fa c to r sh o ck (H )

b

Inv gamma

0.01

2*

0.0027

0.0376

0.0377

0.0332

0.0420

Inve st. te ch . sh o ck (H )

i

Inv gamma

0.01

2*

0.0022

0.0308

0.0309

0.0271

0.0343

D isc o u nt fa c to r sh o ck (F )

b

Inv gamma

0.01

2*

0.0048

0.0501

0.0537

0.0459

0.0612

Inve st. te ch . sh o ck (F )

i

Inv gamma

0.01

2*

0.0030

0.0361

0.0382

0.0331

0.0430

B o rd e r e n fo rc e m e nt sh o ck

fe

Inv gamma

0.01

2*

0.0041

0.0558

0.0557

0.0489

0.0619

Notes: For the Inverted gamma function the degrees of freedom are indicated.

14

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A1. Data and predicted values from the model Home (US) Output

Home (US) Consumption

0.04

0.02

0.02

0.01

0

0

-0.02 -0.04 1982:1

-0.01

Model prediction Data 1985:1

1988:1

1991:1

1994:1

1997:1

2000:1

2003:1

-0.02 1982:1

1985:1

1988:1

Home (US) Investment 0.1

0.2

0.05

0

0

-0.2

-0.05

1985:1

1988:1

1991:1

1994:1

1997:1

1994:1

1997:1

2000:1

2003:1

1997:1

2000:1

2003:1

2000:1

2003:1

Foreign (MEX) Output

0.4

-0.4 1982:1

1991:1

2000:1

2003:1

-0.1 1982:1

1985:1

1988:1

Foreign (MEX) Consumption

1991:1

1994:1

Foreign (MEX) Investment

0.1

0.2 0.1

0.05

0 0

-0.1

-0.05 -0.1 1982:1

-0.2 1985:1

1988:1

1991:1

1994:1

1997:1

2000:1

2003:1

2000:1

2003:1

-0.3 1982:1

1985:1

1988:1

1991:1

1994:1

1997:1

Border Enforcement (hours) 0.2 0.1 0 -0.1 -0.2 1982:1

1985:1

1988:1

1991:1

1994:1

1997:1

Note: Data (solid line) and benchmark model’s Kalman filtered one-sided predicted values (dashed line) for 1981:Q3-2004:Q3. Variables are transformed in Δln, and thus measured in growth rates.

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A2. Autocovariance functions -4

Home/US output (t)

6

-4

x 10

2

4

1

2

2

0

0

0

0

-1

-1

-2

0

1

2

3

4

-2

-4

Foreign/Mex output (t)

-4

x 10

1

2

0

1

2

3

4

10

1

4

0

3

4

0

1

2

3

4

5

0

1

2

3

4

1

2

3

4

1

2

3

4

x 10

0

-5

-5

0

1

2

3

4

-4

0

1

2

3

4

-10

-3

x 10

2

0 -3

x 10

0.15

5

0.1

0

x 10

0

0.05 -2

-1 0

1

2

3

4

-4

-4

x 10

10

-5

0 0

1

2

3

4

-0.05

-4

2

0

1

2

3

4

-10

-3

x 10

5

5

0

0

-5

0 -3

x 10

6

x 10

4

0

2

-2 -4

-4

-4

x 10

-2

0

4

2

x 10

0

1

-2

1

2

5

-1

2

0 -3

x 10

0

-2

-2

-4

x 10

-3

Migration/Appreh. (t)

2

4

-2

Border enforcement (t)

-3

x 10

0

1 2 3 Home/US output (t-h)

4

-5

0

1 2 3 Foreign/Mex output (t-h)

4

-10

0 0

1 2 3 Migration/Appreh. (t-h)

4

-2

0

1 2 3 Border enforcement (t-h)

4

h = 0, 1, ..., 5

Note: The vector auto-covariance function is computed by estimating an unrestricted VAR (1) model with an uninformative prior for the variables plotted. The thin solid line refers to the median vector auto-covariance function, and the dotted lines represent the 2.5 and 97.5 percentiles. The thick solid line refers to the actual data.

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A3. Forecast error variance decompositions Q1

Q4

100

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

0

0 Y

Y*

Technology (H)

C

I

Demand (H)

C*

I*

Technology (F)

Enforc.

Rem.

Demand (F)

Emigr.

Y

Border enf.

Y*

Technology (H)

C

I

Demand (H)

C*

I*

Technology (F)

Q16

Enforc.

Rem.

Demand (F)

Emigr. Border enf.

Q40

100

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

0

0 Y Technology (H)

Y*

C Demand (H)

I

C* Technology (F)

I*

Enforc. Demand (F)

Rem.

Emigr. Border enf.

Y Technology (H)

Y*

C Demand (H)

I

C* Technology (F)

I*

Enforc. Demand (F)

Rem.

Emigr. Border enf.

Note: Forecast variance decomposition at the posterior mode. We include the seven observables in the measurement set, remittances and migration flows. Forecast horizons: Q1, Q4, Q16 and Q40. Technology shocks include both neutral and investment-specific shocks.

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A4. Prior and posterior distributions Std Dev Neutral Tech Shock (Home) 800

Std Dev Neutral Tech Shock (Foreign) 300

600

Std Dev Preference Shock (Home) 150

200

100

100

50

400 200 0

0.01

0.02

0.03

0.04

0.05

0

Std Dev Inv Tech Shock (Home)

0.01

0.02

0.03

0.04

0

0.05

0.01

Std Dev Preference Shock (Foreign)

200

0.02

0.03

0.04

0.05

0.06

Std Dev Inv Tech Shock (Foreign)

150

150

100

100

50

50

150 100 50 0

0.01

0.02

0.03

0.04

0.05

0

Std Dev Border Enforcement Shock

0.02

0.04

0

0.06

gamma (Share of Unskilled in Output, Home)

150

40 30

100

0.02

0.04

0

0.06

eta (Elast of Subst: K and Skilled, Home)

0.05

0.1

0.04

0.05

theta (Elast of Subst: K and Unskilled, Home) 30

0

0.15

zeta (Productivity of Skilled, Home)

0.85

0.9

0.95

1

gamma* (Share of Unskilled in Output, Foreign) 80

40 0.4

30 20

60 40

0.2

20

10 0

0.82 0.84 0.86 0.88

0.9

0.92 0.94

0

theta* (Elast of Subst: K and Unskilled, Foreign)

4

6

8

10

12

0

eta* (Elast of Subst: K and Skilled, Foreign)

0.38

0.4

0.42

zeta* (Productivity of Skilled, Foreign) 0.6

40

40

30

0.4

20

20

0.2

10 0

0.9

0.92

0.94

0.96

0.98

1

0

0.7

fe (Sunk Emigration Cost)

0.75

0.8

1.5

1.5

1

1

0.5

0.5 3

4

2

5

0

4

6

8

mu (Elast Subst Goods, Home) 1.5

2

2

0

psi (Inverse Elast of Labor, Home)

2

0

0.03

10

10

0

0.02

20

20 50

0.01

1 0.5

0.5

1

1.5

2

2.5

0

1

2

3

4

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A4 (continuation). Prior and posterior distributions

nu (Elast of Remittances to Rel. Wages) 4

Persistence, Neutral Tech Shock (Home)

Persistence, Neutral Tech Shock (Foreign)

15

3

10

10

2

5

5

1 0 0.5

1

1.5

0 0.4

0.6

Persistence, Demand Shock (Home)

0.8

0 0.4

1

Persistence, Investment Shock (Home)

0

0.4

0.5

0.6

0.7

0.8

15

10

10

5

5

0

Persistence, Investment Shock (Foreign)

1

20

15

5

0.8

Persistence, Demand Shock (Foreign)

20 10

0.6

0.4

0.5

0.6

0.7

0

0.8

0.4

0.5

0.6

0.7

0.8

Persistence, Border Enforcement Shock

10 100 5

0

50

0.4

0.5

0.6

0.7

0.8

0 0.4

0.6

0.8

1

Note: Benchmark Model. Results based on 500,000 draws of the Metropolis algorithm. Gray line: prior. Black line: posterior. Vertical dashed line: posterior mode (from the numerical optimization of the posterior kernel).

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A5. Impulse response functions to the model’s shocks Neutral Technology Shock in Home -5

6

x 10

Immigrant labor entry

-4

4

4

x 10

Immigrant labor stock

0.2 0.15

2

2

0.1 0

0 -2

Wage immigrant/unskilled, Home

0

5

10

15

20

25

30

Wage unskilled, Foreign

0.01

-2

0.05 0

5

10

15

20

25

30

Remittances per worker

0.015

0

0

5

1.5

0.01

1

0.005

0.5

10

15

20

25

30

25

30

25

30

Total remittances

-3

x 10

0.005

0

0

5

10

15

20

25

30

0

0

5

Output, Home

10

15

20

25

30

0

0.2

0.08

0.8

0.06

0.6

0.1

0.04

0.4

0.05

0.02

0.2

0

0

5

15

20

25

30

Output, Foreign

-4

5

10

x 10

5

0

5 -3

6

0

x 10

10

15

20

25

10

15

20

Capital stock, Home

0.15

0

0

Consumption, Home

30

Consumption, Foreign

0

0

5

0.015

4

0.01

2

0.005

10

15

20

Capital stock, Foreign

-5 -10 -15

0

5

10

15

20

25

30

0

0

5

10

15

20

25

30

0

0

5

10

15

20

25

30

Neutral Technology Shock in Foreign -4

2

x 10

Immigrant labor entry

-4

0

0 -2

-4

0

5

10

15

20

25

30

Wage unskilled, Foreign

0.1

0

5

15

20

25

30

Output, Home

-3

5

10

x 10

Immigrant labor stock

0.08

-2

0.06

-4

0.04

-6

0.02

-8

0

5

10

15

20

25

30

Remittances per worker

0

0.05

0

x 10

0

0 -1

-0.02

-2

-0.03

-3 0

5 -3

6

x 10

10

15

20

25

30

Consumption, Home

-4

4

0.04

-5

2

0.02

0

5

10

15

20

25

30

Output, Foreign

0.08

0

0

10

15

20

25

30

Consumption, Foreign

0.06

0.06

5

0

0

10

15

20

25

30

25

30

25

30

Total remittances

x 10

5

10

15

20

Capital stock, Home

0

5

10

15

20

Capital stock, Foreign

0.4 0.3

0.04

0.04

0.2 0.02

0.02 0

5

0.06

0

-10

0 -3

-0.01

-0.04

Wage immigrant/unskilled, Home

0

5

10

15

20

25

30

0

0.1 0

5

10

15

20

25

30

0

0

5

10

15

20

25

30

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A5 (continuation). Impulse response functions to the model’s shocks Demand Shock in Home -4

2

x 10

Immigrant labor entry

-4

0

1

-1

0

-2

-1

-3

-2

0

5 -3

0

x 10

10

15

20

25

30

Wage unskilled, Foreign

-4

0

-4

-2

-6

-4

0

5

10

15

20

25

30

-6

Immigrant labor stock

0

-0.04

0

5

x 10

10

15

20

25

30

Remittances per worker

-0.06

0

5

10

15

20

25

30

-10

0.4

-0.5

-0.2

0.2

-1

-0.3

0

-1.5

25

30

Output, Foreign

-3

2

20

x 10

-0.2

0

5 -3

0

0

20

25

30

0

5

x 10

10

15

20

25

10

15

20

25

30

25

30

Capital stock, Home

-0.1

15

15

Total remittances

x 10

Consumption, Home 0

10

10

-5

0.6

5

5 -4

5

0

0

0

0

Output, Home

-0.4

Wage immigrant/unskilled, Home

-0.02

-3

2

-2

-8

x 10

30

Consumption, Foreign

-2

0

5

5

10

15

20

x 10

Capital stock, Foreign

0

5

-3

0

-1

-5 -2

-2

-4

-3

0

5

10

15

20

25

30

-10 0

5

10

15

20

25

30

-15

10

15

20

25

30

Investment Shock in Home -4

2

x 10

Immigrant labor entry

-4

4

x 10

Immigrant labor stock

0.06

0

2

0.04

-2

0

0.02

-4

0

5 -3

6

x 10

10

15

20

25

30

Wage unskilled, Foreign

8

2

0

5

10

15

20

25

30

Output, Home

0.4

0

5 -3

4

0

-2

x 10

15

20

25

30

Remittances per worker

0

8 6

4

4

2

2

0

0

5

10

15

20

25

30

Consumption, Home

0

0

1

-0.2

0.5

5

15

20

25

30

Output, Foreign

-3

2

10

x 10

-0.4

0

5 -3

3

x 10

10

15

20

25

30

Consumption, Foreign

0

2

0.01

0

1

0

0

5

10

15

20

25

30

0

15

20

25

30

5

10

15

20

25

30

25

30

Capital stock, Home

0

5

10

15

20

25

0

30

-0.01

5

10

15

20

Capital stock, Foreign

0.02

1

-1

x 10

0

10

Total remittances

1.5

0.1 0

5

2

0.2

0.2

0

0 -4

6

0.4

0.3

10

Wage immigrant/unskilled, Home

0

5

10

15

20

25

30

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A5 (continuation). Impulse response functions to the model’s shocks Demand Shock in Foreign -4

2

x 10

Immigrant labor entry

-3

0

x 10

Immigrant labor stock

0.15 0.1

0 -0.5

0.05

-1

-0.05

-2

-4

Wage immigrant/unskilled, Home

0 0

5

10

15

20

25

30

Wage unskilled, Foreign

0.04

0

5

15

20

25

30

Remittances per worker

0.04

0.02

10

0

5

0.02

0

0

-1

15

20

25

30

25

30

25

30

Total remittances

-3

1

10

x 10

0 -0.02 -0.04

0

5

10

15

20

25

30

-0.02

Output, Home

0

5 -3

0

0

-0.005

x 10

10

15

20

25

30

0

5

Consumption, Home

10

15

20

Capital stock, Home 0

-2

-0.01

-0.05 -4

-0.015 -0.02

-2

0

5

10

15

20

25

30

Output, Foreign

0.1

-6

0

5

10

15

20

25

30

Consumption, Foreign

0.15

-0.1

0.1

-0.1

0

0.05

-0.2

-0.05

0

-0.3

-0.1

-0.05

5

10

15

20

25

30

0

5

10

15

20

25

5

30

-0.4

10

15

20

Capital stock, Foreign

0

0.05

0

0

0

5

10

15

20

25

30

Investment Shock in Foreign -4

5

x 10

Immigrant labor entry

-3

0

0

x 10

Immigrant labor stock

0.2

-0.5

0.15

-5

-1

0.1

-10

-1.5

0.05

-15

0

5

10

15

20

25

30

Wage unskilled, Foreign

0.02

-2

0

5

10

15

20

25

30

Remittances per worker

0.03

0.01

Wage immigrant/unskilled, Home

0

0

5

1

10

15

20

25

30

25

30

25

30

Total remittances

-3

x 10

0

0.02

-1 0

0.01

-0.01

0

0

5

10

15

20

25

30

Output, Home

0

-2 0

5 -3

2

x 10

10

15

20

25

30

Consumption, Home

-3

0

-0.02

-0.02

-2

-0.04

0

5

10

15

20

25

30

Output, Foreign

0.06

0.02

0

5

10

15

20

0

25

30

5

10

15

20

25

30

Consumption, Foreign

0.1

0.04

0

-4

-0.06

0.3

0

0.2

-0.05

0.1 0

5

10

15

20

25

30

0

10

15

20

Capital stock, Home

0

5

10

15

20

Capital stock, Foreign

0.4

0.05

-0.1

5

0

-0.01

-0.03

0

0

5

10

15

20

25

30

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A5 (continuation). Impulse response functions to the model’s shocks Shock to Border Enforcement -4

0

x 10

Immigrant labor entry

-3

0

x 10

Immigrant labor stock

0.4

-2

-1

0.3

-4

-2

0.2

-6

-3

0.1

-8

-4

0

5

10

15

20

25

30

Wage unskilled, Foreign

0

5

10

15

20

25

30

Remittances per worker

0.08

-0.04

0

5

10

15

20

25

30

5

0 -1

0

5

10

15

20

25

30

20

25

30

-2

0

5

10

15

20

25

30

25

30

Capital stock, Home

0

-0.02

15

Total remittances

x 10

Consumption, Home

0

10

1

0.02 0

0 -3

0.04

Output, Home

0.4 0.2

-0.01

-0.04

0 -0.02

-0.06 -0.08

0

2

0.06

-0.02

-0.06

0

Wage immigrant/unskilled, Home

0

5

10

15

20

25

30

Output, Foreign

0.02

-0.03

-0.2 0

5 -3

5

0.01

0

0

-5

-0.01

-10

x 10

10

15

20

25

30

Consumption, Foreign

-0.4

0

5

10

15

20

Capital stock, Foreign

0.08 0.06 0.04

0

5

10

15

20

25

30

0.02 0

5

10

15

20

25

30

0

0

5

10

15

20

25

30

Note: The solid line is the median impulse response to one standard deviation of the shocks; the dotted lines are the 10 and 90 percent posterior intervals.

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A6. MCMC univariate convergence diagnostics gamma (Interval)

0.03

gamma (m2)

-4

1.5

0.025

x 10

gamma (m3)

-6

3

1

2

0.5

1

x 10

0.02 0.015 0.01

2

4

6

8

10

0

2

4

6

8

4

theta (Interval)

0.06

theta (m2)

-4

x 10

2

4

6

8

10 4

x 10

theta (m3)

-5

2

x 10

1.5

4

0.04

1 2

0.03 0.02

0

x 10

6

0.05

10 4

x 10

2

4

6

8

10

0.5

0

2

4

6

8

4

2

4

6

8

x 10

eta (m2)

-5

10

x 10

eta (m3)

-6

1.5

8

10 4

x 10

eta (Interval) 0.02

0

4

x 10

0.025

10

x 10

1

6 0.015 0.01

0.5

4 2

4

6

8

10

2

2

4

6

8

4

10

x 10

4

6

8

10 4

x 10

zeta (m2)

zeta (m3)

1

1

0.8

2

0.6 1.5

0.5

0.4 2

4

6

8

10

0.2

2

4

6

8

4

2

4

6

8

x 10

10 4

x 10

gamma2 (m2)

-5

3

0.013

0

x 10

gamma2 (Interval) 0.014

10 4

x 10

gamma2 (m3)

-7

2.5

x 10

2

2.5

0.012

1.5 2

0.011 0.01

2

x 10

zeta (Interval) 2.5

1

0

4

2

4

6

8

10 4

x 10

1.5

1 2

4

6

8

10 4

x 10

0.5

2

4

6

8

10 4

x 10

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A6 (continuation). MCMC univariate convergence diagnostics theta* (Interval)

0.024 0.022

8

0.02

6

0.018

4

0.016

2

4

6

theta* (m2)

-5

10

8

10

x 10

0.5

2

2

4

6

8

4

eta* (Interval)

10

0

2

4

6

8

4

eta* (m2)

-5

x 10

x 10

eta* (m3)

-6

1.5

8

10 4

x 10

10

0.025

x 10

1

x 10

0.03

theta* (m3)

-6

1.5

x 10

1

6 0.02 0.015

0.5

4 2

4

6

8

10

2

2

4

6

8

4

zeta* (Interval)

0

zeta* (m2)

0.5

0.3

0.2

1

0.2

0.1

0.8

0.1

10

2

4

6

8

4

10

2

6

8

10 4

x 10

fe (m2)

fe (m3)

0.06

0.8

4

x 10

fe (Interval) 1

0.02 0.015

0.04

0.6

0.01 0.02

0.4 0.2

0

4

x 10

10

zeta* (m3)

0.4

1.2

8

8

4

0.3

6

6

x 10

0.4

4

4

x 10

1.4

2

2

4

x 10

1.6

10

2

4

6

8

10

0

0.005 2

4

6

8

4

psi (Interval)

1

0

2

4

6

8

psi (m2)

10 4

x 10

0.06

0.8

10 4

x 10

x 10

psi (m3)

0.03

0.04

0.02

0.02

0.01

0.6 0.4 0.2

2

4

6

8

10

0

2

4

6

8

4

mu (Interval)

1

10

0

mu (m2)

0.2

0.06 0.04

0.4

0.05

0.02

8

10 4

x 10

0

2

4

6

8

10 4

x 10

10

mu (m3)

0.08

0.1

6

8

4

0.15

4

6

x 10

0.6

2

4

x 10

0.8

0.2

2

4

x 10

0

2

4

6

8

10 4

x 10

TECHNICAL APPENDIX – NOT FOR PUBLICATION

Figure A6 (continuation). MCMC univariate convergence diagnostics nu (Interval)

0.4

nu (m2)

0.02

0.3

0.015

3

0.2

0.01

2

0.1

0.005

1

0

2

4

6

8

0

10

2

4

6

nu (m3)

-3

4

8

0

10

4

x 10

2

4

6

8

4

x 10

10 4

x 10

x 10

Note: Univariate convergence diagnostics (Brooks and Gelman, 1998). The first, second and third columns are the criteria based on the eighty percent interval, the second and third moments, respectively. Univariate diagnostics for the shocks are available upon request.

Figure A7. MCMC multivariate convergence diagnostics

Interval

12 10 8 6 4

1

2

3

4

5

6

7

8

9

10 4

x 10

m2

20 15 10 5 0

1

2

3

4

5

6

7

8

9

10 4

x 10

m3

150 100 50 0

1

2

3

4

5

6

7

8

9

10 4

x 10

Note: Multivariate convergence diagnostics (Brooks and Gelman, 1998). The first, second and third graphs are the criteria based on the eighty percent interval, the second and third moments, respectively.

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Immigration, Remittances and Business Cycles SECTION B: REMITTANCES 1

Altruistic Remittances

Our mechanism of remittances described in the paper is isomorphic to the framework of altruistic remittances with heterogeneous remitters in Acosta et al. (2009, henceforth ALM). The ALM framework assumes that established immigrants have limited information regarding capital accumulation, investment in emigration, and labor force participation in the country of origin every period, variables which are chosen by household members residing in Foreign. It also assumes that the amount of remittances is determined one period in advance, and is based solely on a forecast of the immigrant wage in Home and the resident unskilled wage in Foreign, which is the only information available to support workers’decision to remit. Just like in our model, the premise in ALM is that immigrant workers (remitters) know the structure of the Foreign economy absent business cycle ‡uctuations (which are triggered by temporary stochastic shocks). Consequently, in steady state, foreign household members residing in either Home or Foreign enjoy the same amount of consumption per unit of labor, which is equal to cu =lu : Thus, for any immigrant worker j; the steady-state amount of remittances

is equal to the di¤erence between the immigrant wage

and immigrant consumption (expressed in units of the composite good in Home): Q = wi

Qcu : lu

Over the business cycle, ALM also assume that remittances represent an altruistic compensation mechanism between foreign workers residing in Home and Foreign. Worker j’s decision to send remittances is formed at period t 1, and depends on the relative wage forecast computed one period in advance, o n w denoted Ej;t 1 w i;t : In addition to the steady-state level of average remittances , worker j remits an u;t n o w extra lump-sum amount if the forecast is Ej;t 1 w i;t > wwi , case in which j;t = + . A relative wage j;t

u

u;t

forecast above the steady-state ratio signals relative economic hardship for household members residing in

Foreign, and triggers worker j’s decision to remit the extra funds. For symmetry, a relative wage forecast below the steady state results in remittances equal to

: As a result, immigrant worker j bases o n wi;t w the relative wage forecast on a noisy signal that is worker-speci…c: Ej;t 1 w = #j;t 1 w i;t ; where #j;t 1 j;t

=

u;t

u;t

is a random variable drawn from a common uniform distribution U (#j;t 1 ) with support on the interval h i f f f 1 t ; 1 + t ; with 0 < t < 1. From this speci…cation, it follows that E f#j;t 1 g = 1; on average, immigrant workers correctly predict the value of future relative wages. ALM de…ne a threshold value of the forecast signal, #~j;t above their steady-state level, this threshold, #j;t

1

> #~j;t

1:

j;t

1,

so that #~j;t

wi;t 1w

u;t

=

wi wu :

Remittances are

> ; every time remitter j gets a random variable realization above

This property implies a decreasing monotonic relationship between the 1

TECHNICAL APPENDIX - NOT FOR PUBLICATION threshold realization and the actual wage ratio at the time remittances are received: #~i;t where

1

= #~i;t

wi;t 1( w

u;t

);

w #~0 ( w i;t ) u;t

< 0: Moreover, the proportion of immigrant workers that send the extra lump-sum amount of n o #~j;t 1 1+ ft remittances every period is given by: Pr #j;t 1 > #~j;t 1 = 1 . This establishes a decreasing f 2

t

monotonic relationship between average remittances t and the threshold value of the forecast signal #~i;t 1 , R 0 ~ ~ so that t = j;t u(j)dj = (#i;t 1 ) with (#i;t 1 ) < 0: The presence of a continuum of immigrant workers guarantees the di¤erentiability of

t.

Finally, model symmetry guarantees a stationary equilibrium where

= :

t

In essence, the model with heterogeneous remitters in ALM presented here is isomorphic to the mechanism of remittances in the paper. ALM express average remittances as an increasing function of relative wages for any given value of

f t

f,

=

so that

mechanism of remittances in the paper:

t

=%

w

t

= ( w i;t ) with '

wi;t wu;t

u;t

0 ( wi;t ) wu;t

> 0. This result is similar to the

; where ' > 0 is the elasticity of remittances per

worker with respect to relative wages. In context of the ALM framework, the elasticity ' depends on the amount of lump-sum funds, ; and thus characterizes the thrust of the altruistic motive. It also varies with f t;

which can be interpreted as a measure of uncertainty. This link between ' and @ 2 Prfg ~ f @ #@ t

is underpinned by

1

> 0. The intuition is that average remittances are less sensitive to n o w changes in the wage ratio under higher uncertainty about the reliability of the forecast Et 1 w i;t :

the second di¤erential,

=

f t

2(

f 2 t)

u;t

2 2.1

Sensitivity Analysis Bargaining Power and Remittances

We do not analyze non-cooperative settings in which the choice remittances must be sustainable in a dynamic framework (for instance, if remittances are too high, migrants may choose the break up the ties with its household and so on). Instead, our framework must be interpreted as an altruistic one (see Xu, 2007, for a survey on the classi…cation of intra-household models). However, the relative allocation of household consumption between immigrants in Home and residents in Foreign may be interpreted as a proxy for the bargaining power of each group. Here we relax the assumption that members of the foreign unskilled household, either immigrants or residents, have the same consumption per unit of labor in steadystate, cu =lu : Figure B1 shows that the cyclical behavior of remittances holds for the entire range of values of the resident-to-emigrant consumption ratio. Thus, the bargaining power of migrants vs. residents does not alter the cyclical properties of remittances, which are procyclical with home output, countercyclical with the foreign output, and procyclical with the GDP ratio. In general, remittances become slightly more countercylical with output in the country of origin (Foreign) for lower values of the resident-to-migrant consumption ratio (and thus for lower steady state values of remittances). This result is consistent with 2

TECHNICAL APPENDIX - NOT FOR PUBLICATION

the insurance assumption, since a smaller amount of remittances each period implies less consumption smoothing, and therefore triggers larger responses of remittances for negative shocks in the country of origin. This hypothesis is further analyzed below.

2.2

Insurance Properties of Remittances

Absent the possibility of international trade in bonds, the unskilled households can self-insure by either accumulating domestic capital or investing in emigration. In turn, they can deplete the stock of capital not only to consume, but also to invest in migration when the opportunity arises (i.e. we allow for a portfolio choice of capital and migration). One established fact in the migration literature is that households that are liquidity constrained (i.e. the low-income households) do not have resources to …nance emigration, and thus are forced to give up the potential wage gains from moving abroad. See for instance, Chiquiar and Hanson (2005) and Orrenius and Zavodny (2005) for related evidence. In addition, the literature highlights that when the possibility of migrating is constrained, remittances play an important compensation role for the households members left behind (Jasso and Roszenweig, 2010).1 In line with these empirical evidence, we illustrate the relevance of our mechanism of remittances by computing the unconditional correlations of remittances generated by the model for di¤erent values for the share of unskilled in foreign capital (1

k ), using that Kt = (

k)

1 k

1

k

Ks;t

k

+ (1

k)

1 k

Ku;t

1

k

k k

k

Figure B2 shows that when the unskilled households own a very small share of total capital, remittances become more reactive to the business cycles in Foreign, increasing by more when output falls and viceversa. This result is driven by the larger wage di¤erentials in place when unskilled households lack capital as a resource to …nance migration. Consistent with the empirical studies discussed above, remittances in our model provide more insurance when other instruments for consumption smoothing are not in place. To further illustrate this concept, Figure B3 shows the impulse responses to a negative neutral technology (TFP) shock in Foreign that deteriorates the stance of unskilled households. We depict two scenarios, one with low (1

k

= 0:10) and the other with high capital shares (1

k

= 0:60) for the foreign unskilled

households. When the foreign unskilled households own little capital, they have limited resources available to invest in emigration. In responses to the negative TFP shock, even though the unskilled households disinvest heavily from capital (solid line, …rst row), there is not much they can get by depleting the relative small stock of capital. As investment in migration is constrained, unskilled emigration is weaker, and a larger share of unskilled labor are forced to remain in Foreign. In turn, the immigrant unskilled wage in Home falls by less, while the unskilled wage in Foreign falls by more than in the case with large capital 1 More generally, in the macroeconomic literature, the presence of uninsured rule-of-thumb consumers (or "hand-to-mouth"), that neither save nor accumulate capital, is useful to account for the sizable response of aggregate variables (i.e. consumption) to macroeconomic shocks and the lack of consumption smoothing (see Campbell and Mankiw, 1989).

3

1

:

TECHNICAL APPENDIX - NOT FOR PUBLICATION

holdings by the unskilled. As the wage gap widens by more, the increase in remittances is larger following the bad shock. Since productivity shocks in Foreign explain most of the variance of remittances at all horizons (see Appendix A for details), the impulse responses explain why remittances tend to be more countercylical when recipient households have relatively low capital holdings. This also supports the idea that remittances provide more insurance when other instruments for consumption smoothing are not in place.

2.3

Remittances as a Function of the GDP ratio

In the baseline model, remittances are a function of the immigrant-to-resident wage ratio. We parametrize the model to match the data on remittances and the unskilled wage di¤erential between the U.S. and Mexico. Instead, in this extension, we allow for altruistic remittances to be a function of the output di¤erential between Home and Foreign (rather than the wage di¤erential), while maintaining the framework in Acosta et al. (2009). We estimate the elasticity of remittances with respect to the GDP di¤erential between the U.S. and Mexico using quarterly data over 1995:Q1-2006:Q4 and obtain ' = 3. As expected, the framework with remittances as a function of the output di¤erential generates a strongly procylical remittances with respect to the output ratio. Namely, the correlations of remittances with home output, foreign output and their ratio are 0:25,

0:85, and 0:98; respectively. For comparison, when

remittances are a function of the wage di¤erential as in the main paper, the correlation of remittances with the output ratio is signi…cantly lower (0:28): Although wages tend to be procyclical, their correlation with output declines when other shocks (demand, investment and border enforcement) are incorporated into the model. In any case, the insurance implications of remittances hold in both cases.

References [1] Acosta, P., Lartey, E. and Mandelman, F. (2009). "Remittances and the Dutch Disease," Journal of International Economics, 79(1): 102–116. [2] Campbell, J., Mankiw, G. (1989). Consumption, Income and Interest Rates: Reinterpreting the Time Series Evidence. NBER Macroeconomics Annual 1989 (4): 185-246. [3] Chiquiar, D., Hanson, G. (2005). "International Migration, Self-Selection, and the Distribution of Wages: Evidence from Mexico and the United States," Journal of Political Economy, 113(2): 239-281. [4] Jasso, G., Rosenzweig, M. (2010). Remit or Reunify? US Immigrant Parents, Remittances and the Sponsorship of Children. Yale University, manuscript.

4

TECHNICAL APPENDIX - NOT FOR PUBLICATION

[5] Orrenius, P., Zavodny, M. (2005). Self-selection among undocumented immigrants from Mexico. Journal of Development Economics, 78(1):215-240. [6] Xu, Z. (2007). "A survey on intra-household models and evidence," MPRA Paper 3763, University Library of Munich, Germany.

5

TECHNICAL APPENDIX-NOT FOR PUBLICATION Figure B1. Sensitivity of remittances to the resident-to-immigrant consumption ratio 0.6

0.4

Correlations

0.2

0.0

-0.2

-0.4

Correl (remittances, GDP ratio) Correl (remittances, GDP Foreign)

Correl (remittances, GDP Home)

-0.6 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Resident-to-emigrant consumption ratio in Foreign

1.8

2.0

TECHNICAL APPENDIX-NOT FOR PUBLICATION Figure B2. Sensitivity of remittances to the share of unskilled households in foreign capital 0.6

0.4

Correlations

0.2

0.0

-0.2

-0.4 Correl (remittances, GDP ratio) Correl (remittances, GDP Foreign)

Correl (remittances, GDP Home)

-0.6 0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Share of unskilled households in foreign capital

0.50

0.55

TECHNICAL APPENDIX-NOT FOR PUBLICATION Figure B3. Negative Productivity shock in Foreign. Low vs. high capital share for the unskilled.

Immigrant labor entry (Le)

Immigrant labor stock (Li)

2

Unskilled Investment in Foreign

0.5

10

0.4

0

1

0.3

-10

0.5

0.2

-20

0

0.1

-30

Low Capital Share for Unskilled High Capital Share for the Unskilled

1.5

-0.5

0

5

10

15

20

25

30

35

40

0

0

5

Skilled Investment in Foreign

10

15

20

25

30

35

40

-40

0

0

-0.1

-0.05 -0.1

-2

-0.3

-0.15

-4

-0.4

-0.2

-0.5

-0.25

-6

5

10

15

20

25

30

35

40

-0.6

-0.3

-0.7

-0.35

0

5

10

Wage of the unskilled in Foreign

15

20

25

30

35

40

0

5

10

Remittances per worker

0

15

20

25

30

35

40

15

20

25

30

35

40

30

35

40

Trade Balance

0.8

1.4 1.2

-0.2

0.6

1

-0.4

0.8 0.4

0.6

-0.6

0.4

0.2

-0.8 -1

10

Wage of immigrant labor (wi)

0

-0.2

0

5

Consumption Unskilled in Foreign

2

-8

0

0.2 0

5

10

15

20

25

30

35

40

0

0

5

10

15

20

25

30

Note: Impulse Response at the estimated median of the parameters.

35

40

0

0

5

10

15

20

25

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Immigration, Remittances and Business Cycles SECTION C: BOND TRADING BY UNSKILLED 1

Alternative Model with Bond Trading by Skilled and Unskilled

In the baseline model presented in the paper, only the skilled households in each economy trade bonds internationally. In this extension, we allow for both types of households (skilled and unskilled) in each economy to trade bonds, while the foreign unskilled household also invests in labor migration. Bond Holdings We now distinguish among the bond holdings of the representative skilled and unskilled households as follows: In Home, bsh;t+1 and bsf;t+1 are the bond holdings (i.e. home and foreignspeci…c bonds) of the skilled household; buh;t+1 and buf;t+1 are the bond holdings of the unskilled household. u u s s are the and bf;t+1 are the bond holdings of the skilled household; bh;t+1 and bf;t+1 In Foreign, bh;t+1

bonds holdings of the unskilled household. The new market clearing condition for home bonds is sbsh;t+1 + (1

s + (1 s)buh;t+1 + s bh;t+1

bonds is: sbsf;t+1 + (1

u = 0. Similarly, the new market clearing condition for foreign s )bh;t+1

s + (1 s)buf;t+1 + s bf;t+1

u = 0: The current account equation is the s )bf;t+1

same as in the baseline model, but now uses the new de…nitions for aggregate bond holdings in Home: Bh;t+1 = sbsh;t+1 + (1

s)buh;t+1 and Bf;t+1 = sbsf;t+1 + (1

s)buh;t+1 . The mechanism of remittances does

not change. Euler Equations for Bonds in Home Using subscript j 2 fs; ug to denote the household type (skilled and unskilled), the budget constraint for each type of household becomes: wj;t lj;t + rj;t kj;t + 1 + rtb bjh;t + 1 + rtb > cj;t + ij;t + bjh;t+1 +

2

bjh;t+1

2

Qt bjf;t + Tj;t

+ Qt bjf;t+1 +

2

Qt bjf;t+1

2

:

(1)

The Euler equations for home and foreign bonds are: b 1 + bjh;t = (1 + rt+1 )Et

&j;t+1 &j;t

b and 1 + bjf;t = (1 + rt+1 )Et

1

Qt+1 &j;t+1 : Qt &j;t

(2)

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Euler Equations for Bonds in Foreign The budget constraint of the skilled household is similar: s ws;t ls;t + rs;t ks;t + 1 + rtb bh;t =Qt + 1 + rtb s > cs;t + is;t + bh;t+1 =Qt +

2

s bh;t+1

2

s bf;t + Ts;t

s =Qt + bf;t+1 +

2

2

s bf;t+1

:

(3)

For the unskilled household in Foreign, which invests in labor migration in addition to trading bonds, the budget constraint is: wu;t lu;t

u li;t + wi;t Qt 1 li;t + ru;t ku;t + 1 + rtb bh;t =Qt + 1 + rtb

u > cu;t + fe;t wi;t Qt 1 le;t + iu;t + bh;t+1 =Qt +

2

u bh;t+1

2

u bf;t + Tu;t

u =Qt + bf;t+1 +

u bf;t+1

2

2

:

(4)

For each type of household j 2 fs; ug in Foreign, the Euler equations for bonds are: j b = (1 + rt+1 )Et 1 + bh;t

2

"

Qt &j;t+1 Qt+1 &j;t

#

j b = (1 + rt+1 )Et and 1 + bf;t

"

# &j;t+1 : &j;t

(5)

Implications of Financial Integration for Unskilled

One of the main ideas in the paper is that migration and remittances serve as a substitute for cross-border …nancial ‡ows to diversify away from country-speci…c risk. In other words, agents would be reluctant to pay a sunk cost and invest in migration if instead they have the option to lend o¤shore in a frictionless market. In the alternative model in which the unskilled are …nancially integrated, we …nd that migration is not responsive to the business cycle. In Table C1, the unconditional correlation of migration ‡ows with the Home-Foreign output ratio is 0.02. The incentive to migrate still exists when a neutral technology (TFP) shock hits the destination economy, as the resulting increase in productivity boosts the gains from emigration. In Table C2, the conditional correlations of both labor migration and remittances (i.e. conditional on TFP shocks in Home and Foreign) are similar to the ones in the baseline model with the unskilled in …nancial autarky. Also, in Figure C1, a positive TFP shock in Home triggers a positive response from labor migration when the foreign unskilled households are either in …nancial autarky or in …nancial integration. However, the response of labor migration is di¤erent following demand or investment-speci…c shocks in the destination economy. Refer to Figures C2 and C3. In this case, the …nancially integrated households …nd it optimal to reduce the investment in labor migration, and instead use these resources to lend o¤shore, to allow for more consumption or investment demand in Home. Lending o¤shore o¤ers higher returns than investing in migration, even as home output increases (i.e. demand-driven expansion) and the immigrant 2

TECHNICAL APPENDIX - NOT FOR PUBLICATION

wage increases somewhat. This result explains why the unconditional correlation of migration with the home output is negative in the alternative model.

3

Model Fit and Empirical Relevance for Baseline vs.

Alternative

Model The assumption of …nancial autarky for the unskilled (used in the baseline model) is supported by the evidence. For instance, Hohanan (2007) …nds that only 25% of the adult population in Mexico utilize any type of banking services. This number is well below not only that observed in advances economies, but also the values for other emerging countries (for instance, 60% for Chile). The same study also …nds that the lack of access to banking services is positively correlated with poverty. In addition, the fraction of adult population without access to banking services in Mexico is close to the share of unskilled in the Mexican labor force (less than high school degree, 72%). Arguably, if access to banking services is limited, international bond trading is even more restricted. The next step is to …nd out whether the benchmark model (with the unskilled in …nancial autarky) provides a better …t to the data than the alternative model with full …nancial integration. As explained in Section A of this appendix, the marginal likelihood may be interpreted as a summary statistic to assess the out-of-sample performance of the model. The (log) Bayes factor is directly related to the prediction of each model, and is computed as the di¤erence of the log marginal likelihood of each speci…cation. We implement the Modi…ed Harmonic Mean estimator as in Geweke (1999), and consistent with our conjecture, the marginal likelihood di¤erence between the benchmark model and the alternative model is 27.7. This di¤erence indicates a better …t for the baseline speci…cation (see Rabanal and Rubio-Ramírez, 2005, for an explanation).

References [1] Geweke, J. (1999). Using Simulation Methods for Bayesian Econometric Models: Inference Development and Communication. Econometric Reviews 18(1), 1.126. [2] Honohan, P. (2007). Cross-Country Variation in Household Access to Financial Services. The World Bank, manuscript. [3] Rabanal, P., Rubio-Ramírez, J. (2005). Comparing New Keynesian Models of the Business Cycle: A Bayesian Approach. Journal of Monetary Economics, 52(6):1151-1166.

3

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Table C1. Unconditional moments with …nancial integration for unskilled households Theoretical Moments, Unconditional Variable (growth)

St. dev.

Autocorr.

Corr with

GDPh Q GDPf

Corr with GDP h

Corr with Q

Migration ‡ows

15.36

-0.09

0.02

-0.28

-0.23

Remittances

1.80

-0.01

0.54

0.44

-0.33

Border enforcement

5.11

0.00

0.01

0.00

-0.01

Trade balance, Foreign

2.31

-0.37

0.63

0.36

-0.50

GDP f

Note: We simulate the model using the posterior median of the estimated parameters. The unskilled households are …nancially integrated in a frictionless market. Table C2. Conditional moments with …nancial integration for unskilled households Theoretical Moments, Conditional on TFP Shocks Variable (growth)

St. dev.

Autocorr.

Corr with

GDPh Q GDPf

Corr with GDP h

Corr with Q

Migration ‡ows

8.51

-0.07

0.26

-0.02

-0.28

Remittances

1.53

0.01

0.98

0.31

-0.81

Border enforcement

5.11

0.00

0.02

0.01

-0.02

Trade balance, Foreign

0.28

0.24

-0.85

0.05

0.90

Note: We turn o¤ the demand and investment-speci…c shocks ( b ; note to Table C1.

4

i;

b

;

i

= 0). In addition, see the

GDP f

TECHNICAL APPENDIX-NOT FOR PUBLICATION

Figure C1. Impulse Responses to a neutral (TFP) shock in Home, Unskilled Integrated vs. Unskilled in Autarky. Immigrant labor entry (Le)

Immigrant labor stock (Li)

0.8

Home Output

0.4 Autarky Financial Integration

0.6

1.4 1.2

0.3

1 0.4

0.2 0.8

0.2

0.1

0

0

0

5

10

15

20

0.6 0

5

10

15

20

0.4

0

5

Wage of immigrant labor (wi)

Capital in Foreign 0.1 0.08 0.06

10

15

20

15

20

15

20

Consumption in Foreign

1.2

0.17

1

0.16

0.8

0.15

0.04 0.6

0.14

0

0.4

0.13

-0.02

0.2

0.02

0

5

10

15

20

0

5

Remittances per worker

10

15

20

0

5

Trade Balance

1

10 Total Remittances

0.5

0.8

1 0.9

0

0.8

0.6 -0.5

0.7

0.4

0.6 -1

0.2 0

0.12

0

5

10

15

20

-1.5

0.5 0

5

10

15

20

0.4

0

5

10

Note: Impulse responses at the median of the estimated parameters, in the model with financial integration we consider a counterfactual in which the unskilled also trade bonds internationally.

Figure C2. Impulse Responses to an Investment-Specific shock in Home, Unskilled Integrated vs. Unskilled Autarky. Immigrant labor entry (Le)

Immigrant labor stock (Li)

1

Home Output

0.1

0

0

-1

-0.1

-2

-0.2

0.4 0.3 0.2

Autarky Financial Integration

-3 -4

0

5

10

15

0.1

-0.3 20

-0.4

0

5

10

15

20

0

Capital in Foreign 0.4

0.04

0

0.3

0.02

-0.05

0.2

0

-0.1

0.1

-0.02

0

5

10

15

20

0

0

5

Remittances per worker

10

15

20

-0.04

0

5

Trade Balance

0.5

10

15

20

10

15

20

15

20

Total Remittances

2

0.4

0

0.4

0.3

-2

0.3

-4 0.2

0.2

-6

0.1 0

5

Consumption in Foreign

0.05

-0.15

0

Wage of immigrant labor (wi)

0.1

-8 0

Notes: See notes in Figure C1.

5

10

15

20

-10

0

5

10

15

20

0

0

5

10

TECHNICAL APPENDIX-NOT FOR PUBLICATION

Figure C3. Impulse Response to a positive Demand Shock in Home (Negative shock to the discount factor), Unskilled Integrated vs. Unskilled in Autarky. Immigrant labor entry (Le)

Immigrant labor stock (Li)

0.2

Home Output

0.04 Autarky Financial Integration

0.15

0.4

0.03

0.1

0.02

0.05

0.01

0

0

0.3 0.2

-0.05

0

5

10

15

20

-0.01

0.1

0

5

10

15

20

0

0

5

Wage of immigrant labor (wi)

Capital in Foreign 0.02

15

20

15

20

15

20

Consumption in Foreign

0.14

0.025

0.12

0.015

10

0.02

0.1 0.01

0.08

0.015

0.06 0.005 0

0.01

0.04 0

5

10

15

20

0.02

0

5

Remittances per worker

10

15

20

0.005

0

5

Trade Balance

0.1 0.08

10 Total Remittances

0.4

0.1

0.2

0.08

0

0.06

-0.2

0.04

0.06 0.04 0.02 0

0

Note: See Notes to Figure C1.

5

10

15

20

-0.4

0

5

10

15

20

0.02

0

5

10

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Immigration, Remittances and Business Cycles SECTION D: MODEL WITH SOCIAL PLANNER 1

Alternative Model with Social Planner

In the baseline model presented in the paper, the two representative households in each economy solve their optimization problem independently from each other. In this section, we present an alternative model in which a social planner solves the joint optimization problem of the two representative households in each economy, as in Mandelman and Zlate (2010). For comparison, we maintain the assumption of …nancial autarky as in our original model, but introduce two types of labor (skilled and unskilled) in each country, with capital-skill complementarity in production. To save space, we use the same notation as in the baseline model, and restrict our discussion only to the new variables, parameters and equations. The Home Economy The social planner maximizes the weighted sum of utilities of the two representative households (skilled and unskilled):

max

fcs;t ;ls;t ;cu;t ;lu;t ;It ;Kt+1 g

where

and 1

1 X

Et

=t

t

f sU (cs; ; ls; ) + (1

) (1

s) U (cu; ; lu; )g ;

(1)

are the weights of the utility of skilled and unskilled households in the objective function

of the planner. For j fs; ug ; U (cj;t ; lj;t ) are the per-period utilities of the skilled and unskilled households, which take the log-CRRA form with preference shock "bt as in the baseline model. The aggregate budget constraint is: ws;t Ls;t + wu;t Lu;t + rt Kt > Cs;t + Cu;t + It ;

(2)

where capital Kt is aggregate rather than household-speci…c; rt is the gross rental rate of capital. The accumulation of aggregate capital follows the rule: Kt+1 = (1

) Kt + "It It ; where "It is the investment-

speci…c technology shock. The …rst order conditions for capital and labor are: " 1 &t+1 = Et I &t "t where &t =

"bt cs;t

=

(1

)"bt cu;t

1 rt+1 + I "t+1

!#

; and

wj;t = cj;t

j

(lj;t ) ; for j fs; ug ,

(3)

is the aggregate budget constraint multiplier, which is common for the skilled

and unskilled households.

1

TECHNICAL APPENDIX - NOT FOR PUBLICATION

As in the baseline model, production of the home good is a nested CES aggregate of capital, skilled and unskilled labor, with similar expressions for rt , ws;t and wu;t = wi;t . The home composite and the resource constraint also do not change. In Foreign, the social planner maximizes the weighted sum of utilities of

The Foreign Economy

the two representative households: max

fcs;t ;ls;t ;cu;t ;lu;t ;It ;Kt+1 g

where

and 1

1 X

Et

t

s U (cs; ; ls; ) + (1

) (1

s ) U (cu; ; lu; ) ;

(4)

=t

are the weights of the utility of skilled and unskilled households in the objective

function of the planner. For j fs; ug ; U (cj;t ; lj;t ) are the per-period utilities of the skilled and unskilled households as in the baseline model. The aggregate budget constraint is: ws;t Ls;t + wu;t Lu;t

Li;t + wi;t Qt 1 Li;t + rt Kt > Cs;t + Cu;t + fe;t wi;t Qt 1 Le;t + It ;

(5)

where Le;t is the aggregate ‡ow of new emigrant labor, Li;t is the aggregate stock of immigrant labor, and is the rate of return, with the law of movement for the stock of migrant labor: Li;t = (1

l

l )(Li;t 1 +Le;t 1 ):

The …rst order conditions for capital, total labor supply, and new emigrant labor are: " & 1 = Et t+1 I &t "t fe;t wi;t Qt

1

=

1 X

[ (1

1 rt+1 + I "t+1 l )]

t

!#

and

& &t

d

Et

=t+1

where &t =

"bt cs;t

=

(1

)"bt cu;t

wj;t = cj;t

lj;t

for j fs; ug ;

;

(6) (7)

is the aggregate budget constraint multiplier.

The production function for the foreign-speci…c good, the factor prices, the foreign composite, and the resource constraint for the foreign economy are the same as in the baseline model. Under …nancial autarky, the balanced current account condition implies: ph;t Yh;t

pf;t Qt Yf;t = Qt

t.

Finally, the mechanism of

remittances is the same as in the baseline model.

2

Model Implications

The model with a social planner implies that skilled and unskilled households in each economy maximize utility jointly. In this case, once we introduce skill heterogeneity in Foreign, the microfoundations for altruistic remittances are rendered inconsistent, since they are constructed on the assumption of a com-

2

TECHNICAL APPENDIX - NOT FOR PUBLICATION

pensation mechanism between homogenous foreign workers that receive di¤erent wages at the destination and in the country of origin. The setup with a social planner and skill heterogeneity in Foreign waters down the insurance role of the remittances, as the pro…ts from migration are shared by the foreign unskilled households (whose members migrate) with the foreign skilled households (who remain in Foreign) through the social planner. Similarly, unskilled households would get some insurance from the labor and capital income of the skilled households. Table D1 reports the parameter estimates for the alternative model with social planner (…nancial autarky and skill heterogeneity in both Home and Foreign), which are similar to those in the original version of this manuscript (Mandelman and Zlate, 2010, with social planner, …nancial autarky and skill heterogeneity only in Home). In addition, Table D2 displays the simulated moments for the two models. The dynamics of labor migration in the alternative model with social planner are similar to those in the original model (Mandelman and Zlate, 2010). As expected, the only notable di¤erence concerns remittances, which become practically acyclical with the home and foreign output. As explained above, the original insurance role of remittances is diluted when the skilled and unskilled workers share the foreign budget constraint, a result which justi…es our decision to remove the social planner from the baseline model in the revised paper.

References [1] Mandelman, F. and Zlate, A., 2010. Immigration, Remittances and Business Cycles. IFDP 998. Federal Reserve Board.

3

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Table D1: Summary of the prior and posterior distributions of the estimated parameters

Prior distribution Description

Density

Mean

Std Dev

Sd (Hess)

Mode

Mean

5%

95%

S h a re o f u n sk ille d in o u tp u t (H )

Gamma

0.06

0.01

0.0316

0.0683

0.0773

0.0622

0.0907

E la st. o f su b st. K , u n sk ille d , (H )

Beta

0.95

0.015

0.0164

0.9490

0.9458

0.9232

0.9736

E la st. o f su b st. K , sk ille d (H )

Beta

0.85

0.015

0.0092

0.9117

0.9075

0.8923

0.9247

P ro d u c t. o f n a tive sk ille d (H )

Gamma

7

0.75

0.7525

7.0092

8.1265

6.9423

9.517

S h a re o f u n sk ille d in o u tp u t (F )

Gamma

0.40

0.01

0.018

0.3900

0.3984

0.3812

0.414

E la st. o f su b st. K , u n sk ille d , (F )

Beta

0.95

0.015

0.0143

0.9700

0.9574

0.9381

0.9775

E la st. o f su b st. K , sk ille d (F )

Beta

0.73

0.015

0.0339

0.7816

0.7619

0.7411

0.7851

P ro d u c t. o f n a tive sk ille d (F )

Gamma

5.2

0.75

0.9424

5.1974

5.1256

3.8247

6.3912

Gamma

2.8

0.30

0.6586

2.9572

4.7361

4.4185

5.0668

Inve rse e la st. o f la b o r su p p ly

Gamma

1

0.2

0.1631

1.0886

1.6829

1.3396

2.1043

E la st. o f su b stitu tio n , g o o d s

Gamma

1.5

0.3

0.2494

1.5377

2.0222

1.4416

2.5633

Gamma

0.99

0.1

0.1003

0.987

0.9933

0.8115

1.1675

a

Beta

0.75

0.1

0.0269

0.9413

0.9362

0.8984

0.9750

a

Beta

0.75

0.1

0.0364

0.9208

0.9416

0.8935

0.9892

D isc o u nt fa c to r sh o ck (H )

b

Beta

0.5

0.05

0.031

0.7423

0.7223

0.6723

0.7815

Inve st. te ch . sh o ck (H )

i

Beta

0.5

0.05

0.0353

0.772

0.7155

0.6626

0.7731

D isc o u nt fa c to r sh o ck (F )

b

Beta

0.5

0.05

0.001

0.7902

0.7862

0.7839

0.7893

Inve st. te ch . sh o ck (F )

i

Beta

0.5

0.05

0.0094

0.7853

0.626

0.5854

0.6538

B o rd e r e n fo rc e m e nt sh o ck

fe

Beta

0.75

0.1

0.0099

0.9744

0.9918

0.9863

0.998

a

Inv gamma

0.01

2*

0.001

0.0065

0.0067

0.0059

0.0075

a

Inv gamma

0.01

2*

0.0013

0.0178

0.0176

0.0154

0.0196

D isc o u nt fa c to r sh o ck (H )

b

Inv gamma

0.01

2*

0.0032

0.0401

0.0387

0.0346

0.0431

Inve st. te ch . sh o ck (H )

i

Inv gamma

0.01

2*

0.0027

0.0328

0.0326

0.0285

0.0365

D isc o u nt fa c to r sh o ck (F )

b

Inv gamma

0.01

2*

0.0199

0.074

0.0368

0.0311

0.0422

Inve st. te ch . sh o ck (F )

i

Inv gamma

0.01

2*

0.0209

0.0604

0.0256

0.0209

0.0297

B o rd e r e n fo rc e m e nt sh o ck

fe

Inv gamma

0.01

2*

0.0036

0.0507

0.0517

0.0461

0.0577

S u n k e m ig ra tio n c o st

E la st. o f re m itta n c e s to w a g e s

N e u tra l te ch . sh o ck (H ) N e u tra l te ch . sh o ck (F )

N e u tra l te ch . sh o ck (H ) N e u tra l te ch . sh o ck (F )

Name

Posterior distribution

fe

'

4

TECHNICAL APPENDIX - NOT FOR PUBLICATION

Table D2: Unconditional moments with social planner, …nancial autarky (a) Mandelman and Zlate (2010): Skill Heterogeneity in Home Only Variable (growth)

St. dev.

17:57

Migration ‡ows

(13:37=22:27)

1:83

Remittances

(1:61=2:04)

5:18

Border enforcement

(4:67=5:70)

Autocorr.

Corr with

0:15

0:27

(0:22=0:30)

( 0:18= 0:10)

0:06

0:57

( 0:02=0:18)

(0:43=0:72)

0:00

( 0:01=0:00)

GDPh Q GDPf

0:04 ( 0:05= 0:02)

Corr with GDP h

0:03 ( 0:10=0:02)

0:42

(0:31=0:52)

0:01

(0:00=0:01)

Corr with Q

GDP f

0:32 ( 0:36= 0:27)

0:22 ( 0:42= 0:06)

0:05

(0:03=0:06)

(b) Revised Model with Social Planner: Skill Heterogeneity in Home and Foreign Variable (growth)

St. dev.

Migration ‡ows

29:80

(17:76=43:48)

Remittances

2:51

(1:84=3:36)

Border enforcement

5:19

(4:71=5:69)

Autocorr.

Corr with

0:12

0:52

(0:47=0:57)

( 0:16= 0:10)

0:22

(0:08=0:36)

0:04 ( 0:23=0:07)

0:00

(0:00=0:00)

GDPh Q GDPf

0:01 ( 0:02= 0:01)

Corr with GDP h

0:07

(0:01=:12)

0:04

( 0:10=0:18)

0:00

( 0:01=0:00)

Corr with Q

GDP f

0:43 ( 0:51= 0:34)

0:07

( 0:02=0:17)

0:01

(0:00=0:02)

Note: We report the medians from the simulated distribution of moments, using the samples of moments generated with parameters draws from the posterior distribution, for the variables in growth rates. The 5th and 95th percentiles are included.

5

Immigration, Remittances and Business Cycles

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The relationship between business cycles and migration - Empirical Economics Letters 11(1).pdf. The relationship between business cycles and migration ...

Import protection, business cycles, and exchange rates ...
a Development Research Group, Trade and International Integration (DECTI), The World ... We then apply this pre-Great Recession empirical model to realized ...

China's Emergence in the World Economy and Business Cycles in ...
Step 2: Solution to the global model. • Collect all the endogenous variables in a global vector. • Solve simultaneously using the link matrix of country specific.

The uncertainty multiplier and business cycles
Mar 2, 2017 - At the end of the recession, agents' estimates about the extent of recovery are noisy, slowing reactions and delaying ..... How do changes in uncertainty about the current efficiency of investment affect agents' decision making? The key

Economics Letters Asymmetric business cycles and ...
Oct 13, 2017 - In the next sections, we lay out a SOE real business cycle. (RBC) model with default that delivers asymmetric business cycles. Crucially, it does so for normally-distributed productivity shocks,. i.e., there is no skewness in the under