Intertemporal Risk-Return Relationship in Housing Markets

Figures & data

Table 1. Summary statistics of monthly housing returns.

	Mean (%)	SD (%)	Min (%)	Max (%)	Obs.
Housing return	0.06	0.55	−2.54	2.12	789

Notes. The returns are calculated as: $R_t=ln (P_t/P_{t-1}$) where $P_t$ is the price index at time t, from 1954: M1 to 2019: M10. The real U.S. national home price index in Shiller (2015) is used for the construction of the housing return.

Figure 1. Conditional variance generated by GARCH-M models.

Note. This figure plots the conditional variance generated by GARCH-M models over the period February 1954 to October 2019.

Table 2. GARCH-M results at the national level.

	Model specifications		${\alpha }_M$ (risk-return)	${\delta }_0$×10^−6	${\delta }_1$	${\delta }_2$	${\delta }_3$	${\delta }_4$	Log likelihood (R-squared)
(1)	Panel A: GARCH(1,1)-M	Normal	17.81*** (6.27)	10.10*** (1.20)	0.50*** (0.08)	0.12* (0.07)			3111.05 (−0.05)
(2)		Student’s t	16.99*** (6.28)	9.82*** (1.39)	0.52*** (0.09)	0.13* (0.08)			3111.85 (−0.05)
(3)		Generalized	17.80*** (6.33)	10.10*** (1.26)	0.50*** (0.08)	0.12* (0.07)			3111.05 (−0.05)
(4)	Panel B: TGARCH(1,1)-M	Normal	21.49*** (6.36)	10.10*** (1.21)	0.57*** (0.12)	0.12* (0.07)	−0.13 (0.13)		3111.59 (−0.04)
(5)		Student’s t	22.02*** (6.34)	9.83*** (1.45)	0.63*** (0.15)	0.12 (0.08)	−0.19 (0.15)		3112.77 (−0.04)
(6)		Generalized	21.66*** (6.39)	10.10*** (1.31)	0.58*** (0.13)	0.12* (0.07)	−0.14 (0.14)		3111.63 (−0.04)
(7)	Panel C: EGARCH(1,1)-M	Normal	22.46*** (7.00)	−5.04 × 10^6*** (0.75 × 10^6)	0.74*** (0.08)	0.58*** (0.07)	0.02 (0.05)		3104.28 (−0.02)
(8)		Student’s t	23.12*** (6.95)	−4.68 × 10^6*** (0.84 × 10^6)	0.75*** (0.09)	0.62*** (0.08)	0.05 (0.06)		3105.77 (−0.01)
(9)		Generalized	22.73*** (7.03)	−4.97 × 10^6*** (0.80 × 10^6)	0.74*** (0.08)	0.59*** (0.07)	0.03 (0.05)		3104.39 (−0.02)
(10)	Panel D: CGARCH(1,1)-M	Normal	18.41*** (6.65)	23.40*** (4.66)	0.94*** (0.02)	0.08* (0.04)	0.37*** (0.08)	−0.16 (0.14)	3127.83 (−0.05)
(11)		Student’s t	18.41*** (6.72)	23.40*** (4.89)	0.94*** (0.02)	0.08* (0.04)	0.37*** (0.08)	−0.16 (0.14)	3127.83 (−0.05)
(12)		Generalized	19.42*** (6.74)	22.90*** (3.75)	0.94*** (0.02)	0.08** (0.04)	0.36*** (0.07)	−0.18 (0.13)	3129.29 (−0.05)

Notes. The table presents coefficient estimates based on Equations (6)–(11) for different versions of GARCH-M models. Numbers in parentheses denote the corresponding standard errors for the coefficients. ***, **, and * denote significance levels at 1%, 5%, and 10%, respectively.

Table 3. Summary statistics of quarterly housing returns.

	Mean (%)	SD (%)	Min (%)	Max (%)	Obs.
Miami	1.11	2.03	−5.56	5.82	89
Chicago	0.79	0.98	−1.84	3.54	89
Portland	0.73	2.06	−4.77	5.53	89
San Jose	1.22	2.52	−4.88	7.06	89
Cincinnati	0.33	0.75	−1.94	2.68	89
New York	0.96	2.15	−4.26	7.44	89
Los Angeles	1.16	2.56	−4.90	9.60	89
San Francisco	1.26	2.28	−4.49	6.31	89

Notes. We use the deflated FHFA housing price index for the construction of the housing returns. The returns are calculated as: $R_t=ln (P_t/P_{t-1}$) where $P_t$ is the price index at time t from 1985:Q1 to 2007:Q2.

Table 4. Intertemporal risk-return relationship across MSAs.

	(1)	(2)	(3)	(4)
Conditional covariance	10.99*** (2.27)	4.11* (2.39)	22.98*** (2.45)	14.68*** (2.66)
Regional fixed effects	No	Yes	No	Yes
Time fixed effects	No	No	Yes	Yes
Adjusted R-squared	0.12%	2.22%	21.75%	23.96%
Observations	18,128	18,128	18,128	18,128

Notes. The housing price index adopted in this table is based on all-transaction price index from FHFA. This table presents the estimation of $R_{i,t}=c+\left(\mathit{Risk} \mathit{Aversion}\right){\sigma }_{im,t}+{\alpha }_i+{\beta }_t+{\epsilon }_{i,t}$ where$R_{i,t}$ is the real return for each regional housing market,${\sigma }_{im,t}$ denotes conditional covariance with the national market, ${\alpha }_i$ is a regional fixed effect, and ${\beta }_t$ is a time fixed effect. Conditional covariance is constructed through the bivariate GARCH model over the period 1985:Q2 to 2007:Q2. Numbers in parentheses denote the corresponding standard errors for the coefficients. ***, **, and * denote significance levels at 1%, 5%, and 10%, respectively.

Table 5. Risk-return analysis based on the FHFA purchase-only index.

	(1)	(2)	(3)	(4)
Conditional covariance	32.46*** (2.82)	50.34*** (3.67)	13.79*** (2.58)	19.25*** (3.59)
Regional fixed effects	No	Yes	No	Yes
Time fixed effects	No	No	Yes	Yes
Adjusted R-squared	2.01%	4.87%	30.65%	33.19%
Observations	6,400	6,400	6,400	6,400

Notes. The housing price index adopted in this table is based on purchase-only index from FHFA. This table presents the estimation of $R_{i,t}=c+\left(\mathit{Risk} \mathit{Aversion}\right){\sigma }_{im,t}+{\alpha }_i+{\beta }_t+{\epsilon }_{i,t}$ where$R_{i,t}$ is the real return for each regional housing market,${\sigma }_{im,t}$ denotes conditional covariance with the national market, ${\alpha }_i$ is a regional fixed effect, and ${\beta }_t$ is a time fixed effect. Conditional covariance is constructed through the bivariate GARCH model over the period 1991:Q2 to 2007:Q2. Numbers in parentheses denote the corresponding standard errors for the coefficients. ***, **, and * denote significance levels at 1%, 5%, and 10%, respectively.

Table 6. Summary statistics of monthly housing returns.

	Mean (%)	SD (%)	Min (%)	Max (%)	Obs.
Panel A: Capital gain
Miami	0.45	0.37	−0.35	1.49	71
Chicago	0.23	0.31	−0.30	1.30	71
Portland	0.50	0.45	−0.51	1.71	71
San Jose	0.43	0.99	−2.14	2.43	71
Cincinnati	0.34	0.34	−0.26	1.61	71
New York	0.21	0.29	−3.16	1.28	71
Los Angeles	0.32	0.41	−0.75	1.74	71
San Francisco	0.50	0.63	−0.85	1.94	71
Panel B: Rental yield
Miami	0.69	0.04	0.63	0.76	71
Chicago	0.75	0.02	0.71	0.77	71
Portland	0.41	0.02	0.39	0.44	71
San Jose	0.27	0.02	0.23	0.30	71
Cincinnati	0.66	0.02	0.63	0.68	71
New York	0.57	0.02	0.54	0.59	71
Los Angeles	0.37	0.01	0.36	0.38	71
San Francisco	0.30	0.02	0.27	0.32	71

Notes. The deflated rent and house price series from Zillow over 2014:M1 to 2019:M12 is used. The capital gain is calculated as: $R_t=\ln \mathrm{ }(P_t/P_{t-1}$) where $P_t$ is the price index at time t. The rental yield is measured by $\frac{{\mathit{Rent}}_t}{P_t}$ (rents divided by housing prices at time t).

Table 7. Intertemporal risk-return relationship across MSAs.

	(1)	(2)	(3)	(4)
Panel A: Capital gain
Conditional covariance	19.60* (11.22)	11.26 (11.20)	30.44*** (10.07)	20.33** (9.66)
Regional fixed effects	No	Yes	No	Yes
Time fixed effects	No	No	Yes	Yes
Adjusted R-squared	0.03%	13.45%	43.55%	57.72%
Panel B: Rental yield
Conditional covariance	5.43 (3.49)	6.25*** (0.79)	0.21 (4.14)	0.26 (0.68)
Regional fixed effects	No	Yes	No	Yes
Time fixed effects	No	No	Yes	Yes
Adjusted R-squared	0.02%	95.59%	1.29%	97.81%
Observations	7,210	7,210	7,210	7,210

Notes. This table presents the estimation of $R_{i,t}=c+\left(\mathit{Risk} \mathit{Aversion}\right){\sigma }_{im,t}+{\alpha }_i+{\beta }_t+{\epsilon }_{i,t}$ where$R_{i,t}$ is the return for each regional housing market in real terms,${\sigma }_{im,t}$ denotes conditional covariance with the national market, ${\alpha }_i$ is a regional fixed effect, and ${\beta }_t$ is a time fixed effect. Conditional covariance is constructed through the bivariate GARCH model over the period 2014:M2 to 2019:M12. Numbers in parentheses denote the corresponding standard errors for the coefficients. ***, **, and * denote significance levels at 1%, 5%, and 10%, respectively.

Table 8. Intertemporal risk-return relationship of housing in census divisions.

Census division	Conditional covariance	Regional and time fixed effects	Adjusted R-squared	Obs.
Panel A: Capital gain
Pacific	88.43 (65.77)	Yes	48.30%	980
Mountain	71.33 (45.50)	Yes	53.83%	840
New England	109.04 (96.83)	Yes	58.38%	420
South Atlantic	82.49*** (27.09)	Yes	66.53%	1,750
Middle Atlantic	86.06 (54.14)	Yes	56.27%	700
East North Central	15.86 (52.18)	Yes	60.62%	910
East South Central	16.34 (75.83)	Yes	65.16%	490
West North Central	0.93 (26.11)	Yes	72.85%	420
West South Central	−18.80 (37.39)	Yes	65.39%	700
Panel B: Rental yield
Pacific	−3.06* (1.70)	Yes	98.74%	980
Mountain	−1.42 (1.76)	Yes	96.53%	840
New England	−4.58 (4.36)	Yes	92.38%	420
South Atlantic	−2.62 (1.64)	Yes	97.42%	1,750
Middle Atlantic	1.72 (7.10)	Yes	78.00%	700
East North Central	0.99 (2.40)	Yes	98.54%	910
East South Central	−15.79 (13.72)	Yes	89.96%	490
West North Central	0.58 (3.97)	Yes	95.33%	420
West South Central	1.43 (2.47)	Yes	97.24%	700

Notes. This table presents the estimation of $R_{i,t}=c+\left(\mathit{Risk} \mathit{Aversion}\right){\sigma }_{im,t}+{\alpha }_i+{\beta }_t+{\epsilon }_{i,t}$ where$R_{i,t}$ is the return for each regional housing market in real terms,${\sigma }_{im,t}$ denotes conditional covariance with the national market, ${\alpha }_i$ is a regional fixed effect, and ${\beta }_t$ is a time fixed effect. The housing portfolio is segmented at the level of census divisions. Numbers in parentheses denote the corresponding standard errors for the coefficients. ***, **, and * denote significance levels at 1%, 5%, and 10%, respectively.

Table 9. Intertemporal risk-return investigation in housing segmentations.

	Conditional covariance	Regional and time fixed effects	Adjusted R-squared	Obs.
Panel A1 (capital gain): Affordability
Affordable	34.34* (20.47)	Yes	61.45%	3,640
Expensive	44.33* (26.35)	Yes	58.48%	3,570
Panel A2 (rental yield): Affordability
Affordable	−1.57 (2.00)	Yes	92.95%	3,640
Expensive	−1.41 (1.32)	Yes	98.23%	3,570
Panel B1 (capital gain): Population
Large	−26.85 (25.97)	Yes	57.74%	3,640
Small	57.72*** (16.80)	Yes	57.84%	3,570
Panel B2 (rental yield): Population
Large	−3.26** (1.45)	Yes	98.67%	3,640
Small	0.67 (1.85)	Yes	96.94%	3,570

Notes. This table presents the estimation of $R_{i,t}=c+\left(\mathit{Risk} \mathit{Aversion}\right){\sigma }_{im,t}+{\alpha }_i+{\beta }_t+{\epsilon }_{i,t}$ where$R_{i,t}$ is the return for each regional housing market in real terms,${\sigma }_{im,t}$ denotes conditional covariance with the national market, ${\alpha }_i$ is a regional fixed effect, and ${\beta }_t$ is a time fixed effect. The housing portfolio is segmented by housing affordability or population. Numbers in parentheses denote the corresponding standard errors for the coefficients. ***, **, and * denote significance levels at 1%, 5%, and 10%, respectively.

Shiller, R. J. (2015). Irrational exuberance (3rd ed.). Princeton University Press.

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