Appendix: “Time to build and the real-options channel of residential investment” Hyunseung Oh∗and Chamna Yoon†

Contents 1 Micro data supplementary analysis 1.1 Definition of variables in table 1 . . . . . . . . . . . . . . . . . . . . . 1.2 Alternative measure of economic time to build . . . . . . . . . . . . . 1.3 Permit to completion . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 4 4

2 Model estimation with lower interest rate

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3 Model extension with explicit 3.1 Solution characterization . . 3.2 Numerical example . . . . . 3.3 Estimation results . . . . . .

6 7 8 8

∗ †

inventories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vanderbilt University. E-mail: [email protected]. Baruch College, City University of New York. E-mail: [email protected].

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1

Micro data supplementary analysis

1.1

Definition of variables in table 1

In this paper, definitions of the variables used in the regression of table 1 are provided. We follow the definitions provided in the Census Bureau’s “Survey of Construction” webpage. For details, refer to their webpage.1 1. Census division states • New England: CT, MA, ME, NH, RI, VT • Middle Atlantic: NJ, NY, PA • East North Central: IL, IN, MI, OH, WI • West North Central: IA, KS, MN, MO, ND, NE, SD • South Atlantic: DC, DE, FL, GA, MD, NC, SC, VA, WV • East South Central: AL, KY, MS, TN • West South Central: AR, LA, OK, TX • Mountain: AZ, CO, ID, MT, NM, NV, UT, WY • Pacific: AK, CA, HI, OR, WA 2. Construction category • Built for sale: All houses built on builder’s land with the intention of selling the house and land in one transactions. Also called “speculatively-built” houses. • Contractor-built: All houses built for owner occupancy on the owner’s land with construction under the supervision of a single general contractor. • Owner-built: All houses built for owner occupancy, on the owner’s land, under the supervision of the owner acting as the general contractor. • Built for rent: All houses built on builder’s land with the intention of renting the housing unit. 3. Construction method 1

https://www.census.gov/construction/chars/definitions/

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• Modular: Finished 3-dimensional sections of the complete dwelling are built in a factory and transported to the site to be joined together on a permanent foundation. • Panelized: A package of wall panels, roof trusses, and other components is shipped from a factory to be assembled on site. This may include all materials required to finish the house as a complete package. • Site built: The house is built entirely on site, except that it may include some factory components as roof and floor trusses, wall panels, doorframes, etc. 4. Stories • 1 story: Basement is not counted as a story even if it is finished. • 2+ story: Includes 1.5 stories, where living accommodations located wholly or partly within the roof frame is considered a half story. 5. Floor area (square foot): Completely finished floor space, including space in basements and attics with finished walls, floors, and ceilings. This does not include a garage, carport, porch, unfinished attic or utility room, or any unfinished area of the basement. 6. Metropolitan area: Whether or not the house is included in the metropolitan area. 7. Number of full bath: A full bathroom is one that has a washbasin, a toilet, and either a bathtub or shower. 8. Detached: Row houses, duplexes, quadruplexes, or townhouses may also defined as attached single-family houses if they (1) are separated by a ground-to-roof wall, (2) have a separate heating system, (3) have individual meters for public utilities, and (4) have no units located above or below. 9. Deck: A deck house has a floored area without a roof, not sitting directly on the ground, typically made of wood or wood products. 10. Parking facility: (1) 1 car garage, (2) 2 car garage, (3) 3 or more car garage, or (4) other 3

11. Foundation: (1) full or partial basement, (2) crawl space, (3) slab, or (4) other (raised supports, earthen, etc) 12. Material of wall (framing material): (1) wood, (2) steel, (3) concrete/masonry other than insulated concrete forms, or (4) insulated concrete forms (or SIP)

1.2

Alternative measure of economic time to build

In this part, we redo the regression in table 1 in the main text with time to build (TTB) regressed in levels instead of logs. The result is reported in table A1. The left panel of Figure A1 compares the kernel density of the alternative economic TTB measures in 2003 and 2005. Since the regression is now conducted in levels, there are scale differences in the range of economic TTB. In particular, a level TTB regression shows a larger range of residuals on the right tail. However, qualitatively, the description in the paper remains. We again observe that the overall distribution seems to have shifted to the right in between these two years. The right panel of Figure A1 compares the kernel density of the alternative economic TTB measures in 2005 and 2009. The mass of the distribution including the mode shifting back to the left, and a fat tail appearing are also a pattern observed with the alternative economic TTB measure. These patterns are not division specific, as observed in Figure A2. A right fat tail in 2009 appears in all 9 divisions. The fat tail to the right appears only for unsold houses, as observed again in Figure A3. In particular, houses that remain unsold for a longer time show even longer TTB in 2009 compared to 2005. Lastly, these alternative measures lead to similar average TTB implications observed in Figure 6 of the main text. Figure A4 shows economic TTB and ex-bottleneck TTB based on a level regression of TTB. Economic TTB increases by 23 percent, and ex-bottleneck TTB incrases by 28 percent compared to 2003.

1.3

Permit to completion

For houses requiring building permits, the survey of construction data also record the permit issue month. Typically, the period from permit to start takes about a month. However, the average permit to start period have also increased to about 2 months in 2009. Therefore, while average completion from start have increased from 4

Table A1: Regression on TTB (T T B in levels) Frequency New England Middle Atlantic East North Central West North Central South Atlantic East South Central West South Central Mountain Pacific Built-for-sale Contractor-built Owner-built Build-for-rent Modular Panelized Site built 1 Story 2+ Story Square feet (×100) Unemployment rate Constant Other controls: Metropolitan area Number of full bath Detached Deck Parking facility Foundation Material of wall Observations

0.034 0.071 0.124 0.076 0.269 0.053 0.144 0.098 0.132 0.740 0.142 0.087 0.032 0.028 0.024 0.948 0.430 0.570

(a)

(b)

1.894 1.983 1.049 0.756 0.238 0.436 0.190 1.357 –1.454 –0.387 2.049 –2.974 –0.834 0.0510 0.0669

(0.0973) (0.0701) (0.0551) (0.0648) (0.0408) (0.0539)

(0.0241) (0.00176)

9.020

(0.120)

(0.0537) (0.0571) (0.0798) (0.0842) (0.0996) (0.0719) (0.0723)

1.784 1.991 1.178 0.545 0.148 0.543 0.0870 1.552 –1.515 –0.421 2.009 –3.020 –0.827 0.0450 0.0672 –0.252 10.41

yes yes yes yes yes yes yes

yes yes yes yes yes yes yes

261,184

261,184

(0.0970) (0.0699) (0.0551) (0.0645) (0.0407) (0.0540) (0.0534) (0.0572) (0.0796) (0.0839) (0.0994) (0.0722) (0.0721)

(0.0240) (0.00175) (0.00604) (0.126)

Note: Robust standard errors in parentheses.

6 months to 8 months in 2009, the average completion from permits have increased from 7 months to 10 months, which is a larger increase in the proportion. In Figure A5, we reestimate all our TTB measures based on permit issuance month instead of start month and plot each measures as a percentage difference from 2003. Consistent with the rough observation, we find that the increase in average permit issuance to

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completion between 2003 and 2009 is even more in percentage than that in average start to completion in the same period. This suggests that overall construction activity may have slowed down in this period. Although further data is needed, we conjecture that the slowdown in construction may not be confined to TTB or time to start a permitted project, but may also extend to the time to convert a lot to a building permit, or the time to purchase a lot. Adding all these time up could be a significant slowdown in the construction industry.

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Model estimation with lower interest rate

For robustness, we re-estimate the baseline model with the annual real interest rate set at 6.5 percent as described in section 5. The estimation results are plotted in Figures A6, A7a, and A7b. Figure A6 shows that the estimation fits the targeted distribution pattern well, in particular the dynamics of each distributional moment across time. Looking into Figure A7a, the two main lessons remain: (1) construction bottleneck increased until 2007 and fell afterwards, (2) uncertainty was high in 2007 and 2009. In Figure A7b, the rise in uncertainty in 2009 contributes to more than 30 percent of the increase in TTB relative to 2003. On the other hand, while uncertainty is estimated to be similarly high in 2007, its contribution to the rise in TTB in 2007 is lower since house prices were high in this period which plays a nonlinear role for the rise in uncertainty to affect TTB investment.

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Model extension with explicit inventories

In this section, we extend the baseline model in section 4 of the main text to include an explicit inventory channel. That is, even after completion, the house is not guaranteed to be sold. We introduce the notion of time-to-sell (TTS) by assuming an additional demand process for the house. We introduce two probabilities for construction: demand probability up to completion, and demand probability after completion. With inventories and time to sell also introduced, the model incorporates an additional channel of TTB variation with fluctuations with demand conditions. Homebuilders start construction not necessarily expecting to face demand in the future, 6

and they might be forced to wait for a certain amount of time for demand to appear. Extending the model from section 4, the terminal condition changes. Assuming that the demand probability for a completed house is s1 , the expected future terminal condition of a house that will be completed at the end of the period is U U U , 0, Bit+1 ; Λ), ) + (1 − s1 )EV un (Pit+1 , 0, Bit+1 ; Λ) = s1 E(P M Pit+1 EV (Pit+1

(1)

where V un (·) is the conditional value function of an unsold completed house. For unsold completed houses, we assume that the demand probability in each period is s2 , so that the value function could be written as V un (PitU ,0, Bit ; Λ)     1 U U = s2 E(P M Pit+1 ) + (1 − s2 )EV un (Pit+1 , 0, Bit+1 ; Λ) . 1+r

(2)

In this model, the frequency of selling a house up to completion is s1 . Also, the mean duration of time to sell for an unsold completed house 1/s2 . For example, when s2 = 0.5, the average duration of an unsold completed house to last in the market is 2 periods. The two parameters could be easily targeted by the data. Our baseline model is nested when s1 = 1.

3.1

Solution characterization

The value of a completed house with demand is linear in the unit-level price PitU . Based on the form of (2), guess that the value of an unsold completed house is V un (PitU , 0, Bit ; Λ) = αP M PitU , and verify the value of α. Plugging in this guess into (2), we obtain the following expression: αP

M

PitU

 =

1 1+r



  s2 P M PitU + (1 − s2 )αP M PitU ,

U where we used the fact that EPit+1 = PitU . Therefore, we get α = s2 /(s2 + r), and

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the value of a completed house without demand is verified as V un (PitU , 0, Bit ; Λ) =

s2 P M PitU . s2 + r

The solution is intuitive. The value of a completed house without current demand is increasing in s2 , the probability of getting demand in the future.

3.2

Numerical example

The period is monthly. Physical TTB is set at 5 months. The parameter r is set as the annual interest rate 10 percent, and σ is set at an annualized value of 0.2, and P M = 2. We experiment with several parameter values for demand probability s1 and s2 . In Figure A8, we plot the TTB investment price threshold relative to the threshold price of starting construction. The line labeled “100% demand” is the baseline model with s1 = s2 = 1. The two other lines labeled “50% demand, TTS 10 months” and “25% demand, TTS 20 months” are investment price thresholds when s1 = 0.5, s2 = 0.10 and s1 = 0.25, s2 = 0.05, respectively. As we see here, TTB investment decisions are more likely to be delayed when demand probability is low. Our numerical example verifies that TTS is an additional plausible channel of the long TTB in the bust period. However, based on the slope differences in policy functions, the low demand probabilities do not seem to translate to significant differences in TTB decisions. We estimate the extended model to verify whether the new demand channel turns out to be quantitatively important.

3.3

Estimation results

The above model introduces two additional parameters s1 and s2 , the demand probabilities for a newly completed house and an unsold house. In our micro-data, we compute the sales probability of a completed house and the 3 month sales probability of an unsold houses at completion (Table A2). Indeed, sales probability was low in 2008-09. The probability of a completed house to be sold was 59 percent, and out of the remaining 41 percent of houses, the probability of them being sold in 3 months was only 8.79 percent. We calibrate our model with these numbers and re-conduct

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the SMM estimation. Figures A9 and A10 plot the results. We find that the low demand probabilities are limited in accounting for the long TTB in these periods. In Table A2, we find that the demand probabilities for completed houses were also low in 2011. However, 2011 was a period where TTB has already become quite short. Table A2: Sales probabilities of completed units Years (a) (b) 2002–03 76.88% 15.58% 2004–05 76.87% 14.74% 2006–07 64.85% 9.70% 2008–09 59.03% 8.79% 2010–11 59.72% 12.39% 2012–13 74.20% 17.80% Note: (a) cumulative sales probability of a newly completed house, including sales that occurred before completion; (b) the 3 month average sales probability of unsold houses at completion.

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List of Figures A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

Main text Figure 3 with alternative economic TTB . . . . Main text Figure 4 with alternative economic TTB . . . . Main text Figure 5 with alternative economic TTB . . . . Main text Figure 6 with alternative economic TTB . . . . Start to complete versus permit to complete . . . . . . . . Main text Figure 9 with an alternative interest rate . . . . Main text Figure 10 with an alternative interest rate . . . Uncertainty-invariant threshold price of investment relative Main text Figure 9 with inventory and time to sell . . . . Main text Figure 10 with inventory and time to sell . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . to start . . . . . . . . . .

. . . . . . . . . .

11 12 13 14 15 16 17 18 19 20

Figure A1: Main text Figure 3 with alternative economic TTB 0.2

0.2 2003 2005

Density

0.18

2005 2009

0.18

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Note: Kernel density of economic TTB for total single-family houses. Left compares 2003 and 2005, and right compares 2005 and 2009.

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Figure A2: Main text Figure 4 with alternative economic TTB

Density

New England

Middle Atlantic 0.24

0.24

0.16

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0.08

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Density

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South Atlantic 0.24

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Pacific

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Mountain

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West South Central

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Density

East North Central

0.24

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Note: Kernel density of economic TTB for total single-family houses.

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Figure A3: Main text Figure 5 with alternative economic TTB Sold before start

Unsold before start

TTB by sales lag 20

0.25

0.25

0.2

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2005 2009

0.15

0.15

0.1

0.1

Average TTB

Density

15

10

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0 2 4 6 8 10 12 14 16 18 20 22 24

Sales month from start

Note: Left two figures are kernel density of economic TTB for built-for-sale houses, for 2005 and 2009. The right figure plots the average economic TTB by sales month from start, for 2005 and 2009. Houses that were sold before start are lumped into 0, and 24 includes all houses that were sold after 24 months or not sold.

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Figure A4: Main text Figure 6 with alternative economic TTB 30

Percent difference from 2003

25

Raw TTB Economic TTB Ex-bottleneck TTB

20

15

10

5

0

-5

-10 2003

2004

2005

2006

2007

2008

14

2009

2010

2011

2012

2013

Figure A5: Start to complete versus permit to complete Economic TTB

Ex-bottleneck TTB

30

30

25

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Percent difference from 2003

30

Percent difference from 2003

Percent difference from 2003

Raw TTB

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Permit to complete Start to complete

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Note: Start to complete raw TTB, economic TTB, and ex-bottleneck TTB are all the same as in Figure 6 in the main text. Permit to complete measures add permit to start periods to their respective start to complete measures.

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Figure A6: Main text Figure 9 with an alternative interest rate -6m 0.8

6

0.6

Density

Month

Mean 8

4 2

0.4 0.2

0

0 2003 2005 2007 2009 2011 2013

2003 2005 2007 2009 2011 2013

10-12m 0.4

0.3

0.3

Density

Density

7-9m 0.4

0.2 0.1

Data Model estimation

0.2 0.1

0

0 2003 2005 2007 2009 2011 2013

2003 2005 2007 2009 2011 2013

Note: TTB distribution (SMM). Annual interest rate is assumed to be 6.5 percent (In the main text, the interest rate is set at 10 percent). Other parameters are set the same as in Figure 9. The 4 moments used in the estimation of parameters (mean TTB, fraction of TTB in 4-6 months, 7-9 months, and 10-12 months) for each odd year are plotted in each panel above. The grey bars are the empirical moments and the black bars are the estimation based moments from SMM.

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Figure A7: Main text Figure 10 with an alternative interest rate (a) Model parameters Difference from 2003 (percent)

40 30 20 10 0 -10 -20 2003

Price Bottleneck Uncertainty

2005

2007

2009

2011

2013

(b) Historical decomposition of TTB Difference from 2003 (percent)

40 30 20

Model estimation Price effect Bottleneck effect Uncertainty effect

10 0 -10 -20 2003

2005

2007

2009

2011

2013

Year

Note: SMM for baseline model. Annual interest rate is assumed to be 6.5 percent (In the main text, the interest rate is set at 10 percent). Other parameters are set the same as in Figure 10. Panel (a) plots the estimated parameters (bottleneck and uncertainty) relative to the first year in percentage terms. The price levels relative to the first year are set based on the two-year average median sales price of new homes sold (Census) deflated by the CPI (BLS).

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Figure A8: Uncertainty-invariant threshold price of investment relative to start

Note: The plotted measure is (1/σ)[log P ∗ (K; σ) − log P ∗ (1; σ)], and the x-axis is 5 × K. For this plot, we assume physical TTB to be 5 months. The annualized value of σ is 0.2.

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Figure A9: Main text Figure 9 with inventory and time to sell Mean

-6m

10

0.8

Density

Month

0.6 5

0.4 0.2

0

0 2003 2005 2007 2009 2011 2013

2003 2005 2007 2009 2011 2013

10-12m 0.4

0.3

0.3

Density

Density

7-9m 0.4

0.2 0.1

Data Model estimation

0.2 0.1

0

0 2003 2005 2007 2009 2011 2013

2003 2005 2007 2009 2011 2013

Note: TTB distribution (SMM). The baseline model is extended to incorporate two demand probabilities: demand for a newly completed house and demand for an unsold completed house. Other parameters are set the same as in Figure 9. The 4 moments used in the estimation of parameters (mean TTB, fraction of TTB in 4-6 months, 7-9 months, and 10-12 months) for each odd year are plotted in each panel above. The grey bars are the empirical moments and the black bars are the estimation based moments from SMM.

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Figure A10: Main text Figure 10 with inventory and time to sell (a) Model parameters Difference from 2003 (percent)

40 30 20 10 0 -10 -20 -30 2003

Price Bottleneck Uncertainty Demand

2005

2007

2009

2011

2013

(b) Historical decomposition of TTB Difference from 2003 (percent)

40 30 20

Model estimation Price effect Bottleneck effect Uncertainty effect Demand effect

10 0 -10 -20 2003

2005

2007

2009

2011

2013

Year

Note: SMM for baseline model. The baseline model is extended to incorporate two demand probabilities: demand for a newly completed house and demand for an unsold completed house. Other parameters are set the same as in Figure 10. Panel (a) plots the estimated parameters (bottleneck and uncertainty) relative to the first year in percentage terms. The price levels relative to the first year are set based on the two-year average median sales price of new homes sold (Census) deflated by the CPI (BLS). The demand probability is depicted only for the sales frequency of newly completed houses.

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