Why the Housing Sector Leads the Whole Economy: The Importance of Collateral Constraints and News Shocks

Abstract

This paper establishes a dynamic stochastic partial equilibrium model for explaining residential investment dynamics in the United States, focusing on the distinctive cyclical features of residential investment in that it leads the whole economy. This paper is different from the existing literature by adding three new features to the model: news shocks, collateral constraints and agent heterogeneity. The partial equilibrium analysis where interest rates are exogenously fixed shows that these assumptions are essential to generating the dynamic pattern in which residential investment leads consumption and GDP.

Appendix

A. Data

This section describes all the data used in this paper.

• 1. Gross domestic product, BEA, NIPA table 1.1.5, line 1

• 2. Personal consumption expenditures, BEA, NIPA table 1.1.5, line 2

• 3. Gross private domestic investment, BEA, NIPA table 1.1.5, line 7

• 4. Government consumption expenditure and gross investment, BEA, NIPA table 1.1.5, line 21

• 5. Residential investment, BEA, NIPA table 1.1.5, line 21, line 12

• 6. Imputed rentals of self-owned houses, BEA, NIPA table 2.5.5, line 21

• 7. Consumer price index, BEA, NIPA table 1.1.4, line 2

• 8. Aggregate house values, BEA, NIPA table 2.1, line 60

• 9. Mortgage rates, contract interest rates on commitments for fixed-rate first mortgages (30 years), Board of Governors of the Federal Reserve System

• 10. Nominal output = 1 − 3 + 5 − 4 − 6. The output consists of the consumption and the residential investment but not business investment. Because the model we use is an endowment economy, we do not consider the business investment. Hence, we exclude the business investment from the subjects of research and similarly government expenditures.

• 11. Nominal consumption = 2–6. The consumption is composed of non-durable goods, durable goods and services. To be consistent with the model, we subtract the imputed rentals of self-owned houses from the consumption.

• 12. The ratio of residential investment to personal consumption expenditure = the mean of 5/11

• 13. The ratio of residential investment to the aggregate house values = the mean of 5/8

• 14. The ratio of aggregate house values to nominal output = the mean of 8/10

• 15. Real output = 10/7

• 16. Real consumption expenditure = 11/7

• 17. Real residential investment = 5/7

All the real data are detrended with the Hodrick–Prescott filter.

B. Discretizing Two-Dimensional AR(1) Process

We extend the application of the method proposed by Tauchen (1986) from one-dimensional AR(1) process into the following two-dimensional AR(1) process defined by the following

$$ \begin{array}{rll} z^\prime &=& \rho z +\nu ,\\ s &=& \rho z+\eta , \end{array} $$

where z is the current productivity, s is the current signal and z ^′ denotes the next-period productivity. The disturbances (ν,η) are i.i.d processes and satisfy the following conditions:

$$ \begin{array}{rll} (\nu,\eta) &\perp& z\\ (\nu,\eta)&\sim& N(0,\Pi), \end{array} $$

where Π is defined by:

$$ \Pi =\sigma^2 \left[ {\begin{array}{cc} 1 & \pi \\ \pi & 1 \\ \end{array} } \right]. $$

Following Tauchen (1986), we assume the members of $\mathcal{Z}$ evenly-spaced and denoted as $\mathcal{Z}=\{z_1,z_2,\dots,z_{n-1},z_n\}$ which satisfies

$$ z_1<z_2<\dots<z_n.\\ $$

The upper and the lower bounds on the range, z ₁ and z _n , respectively, are set to m unconditional standard deviations on each side of 0.⁶ Therefore, z ₁ and z _n satisfy the following

$$ z_1=-m\frac{\sigma}{\sqrt{1-\rho^2}}\\ $$

and

$$ z_n=m\frac{\sigma}{\sqrt{1-\rho^2}}.\\ $$

We also assume the signals $s\in\mathcal{S}=\mathcal{Z}$. If we define the state composed of the current productivity and the signal, i.e. (z,s), the probability transition matrix is determined as follows.

Let ω = z _i  − z _i − 1 (i > 1). Given (z,s), the probability of z ^′ satisfies the following.

If 1 < i < n,

$$ \begin{aligned} &\text{Prob}\left(z^\prime=z_i|z=z_m,s=z_h\right)\\ &\quad=\text{Prob}(z_i-\omega/2\leq\rho z_m+\nu\leq z_i+\omega/2|\eta=z_h-\rho z_m)\\ &\quad=\Phi\left[\frac{z_i+\omega/2-\rho z_m-\rho(z_h-\rho z_m)}{\sigma\sqrt{(1-\rho^2)}}\right]\\ &\qquad - \Phi\left[\frac{z_i-\omega/2-\rho z_m-\rho(z_h-\rho z_m)}{\sigma\sqrt{(1-\rho^2)}}\right] \end{aligned} $$

If i = 1,

$$ \begin{aligned} &\text{Prob}\left(z^\prime=z_i|z=z_m,s=z_h\right)\\ &\quad=\text{Prob}(\rho z_m+\nu\leq z_1+\omega/2|\eta=z_h-\rho z_m)\\ &\quad=\Phi\left[\frac{z_1+\omega/2-\rho z_m-\rho(z_h-\rho z_m)}{\sigma\sqrt{(1-\rho^2)}}\right] \end{aligned} $$

If i = n,

$$ \begin{aligned} &\text{Prob}\left(z^\prime=z_i|z=z_m,s=z_h\right)\\ &\quad=\text{Prob}(\rho z_m+\nu\geq z_n-\omega/2|\eta=z_h-\rho z_m)\\ &\quad=1-\Phi\left[\frac{z_n-\omega/2-\rho z_m-\rho(z_h-\rho z_m)}{\sigma\sqrt{(1-\rho^2)}}\right] \end{aligned} $$

where Φ(·) denotes the cumulated distribution function of standard normal distribution. Also, we know that given z ^′, the probability of s ^′ satisfies the following:

If 1 < j < n,

$$ \begin{aligned} \text{P}\left(s^\prime=z_j|z^\prime=z_i\right)&=\text{P}(z_j-\omega/2\leq\rho z_i+\eta\leq z_j+\omega/2)\\ &=\Phi\left[\frac{z_j+\omega/2-\rho z_i}{\sigma}\right]- \Phi\left[\frac{z_j-\omega/2-\rho z_i}{\sigma}\right]\\ \end{aligned} $$

If j = 1,

$$ \begin{aligned} \text{P}\left(s^\prime=z_1|z^\prime=z_i\right) &=\text{P}\left(\rho z_i+\eta\leq z_1+\omega/2\right)\\ &=\Phi\left[\frac{z_1+\omega/2-\rho z_i}{\sigma}\right] \end{aligned} $$

If j = n,

$$ \begin{aligned} \text{P}(z^\prime=z_n|z^\prime=z_i)&=\text{P}(\rho z_i+\nu\geq z_n-\omega/2)\\ &=1-\Phi\left[\frac{z_n-\omega/2-\rho z_i}{\sigma}\right] \end{aligned} $$

Then the transition probability is defined by

$$ \text{Prob}(z^\prime,s^\prime|z,s)=\text{Prob}(z^\prime|z,s)\text{Prob}(s^\prime|z^\prime). $$

C. Parameters of Income Process

This appendix displays the parameters of the processes of the individual income. The set of states for the individual income process (normalized to 1) is given by

$$ \left( \begin{array}{ccccc} 0.2468& 0.4473& 0.7654& 1.3097& 2.3742\\ \end{array} \right). $$

And the transition matrix Ω is given by

$$ \left( \begin{array}{ccccc} 0.7367 & 0.2473 & 0.0150 & 0.0002 & 0.0000\\ 0.1947 & 0.5555 & 0.2328 & 0.0169 & 0.0001\\ 0.0113 & 0.2221 & 0.5333 & 0.2221 & 0.0113\\ 0.0001 & 0.0169 & 0.2328 & 0.5555 & 0.1947\\ 0.0000 & 0.0002 & 0.0150 & 0.2473 & 0.7376\\ \end{array} \right). $$