Five themes of U.S. home price cycles: a dynamic modelling approach

Abstract

After the Global Financial Crisis, the U.S. housing market has been studied extensively from several dimensions to assess the causes for the price crash during 2007–2012. In this paper, we formulate five hypotheses about the behavior of housing prices and introduce two important innovations: first we extend the sample period, from January 1987 to August 2023, to assess whether the common determinants that drove housing prices during the GFC also moved housing dynamics during the essentially zero interest rate and pandemic periods. Second, we formulate five hypotheses and employ a dynamic econometric modelling approach to empirically investigate the drivers of house prices. The hypotheses proposed address the macroeconomic business cycle environment, monetary policy, the global saving glut, the fundamentals of the housing market, and housing momentum. We find that most of the independent variables in our hypotheses are statistically significant except for momentum. We conclude that several economic variables have driven the housing market during the past 36 years but with differing impact across sub-periods.

Appendices

Appendix:

General form of dynamic parameter model and estimation of the model parameters

We describe briefly how the unknown parameters in the dynamic model may be estimated. Our aim is to present an overview of the filtering and smoothing algorithm (known as Kalman filter and smoother) and the optimization of the likelihood function. Before proceeding, however, it is advantageous to express the dynamic model in term of suitable notations. This is advantageous since the discussion here is applicable to any such state space formulation and not restricted to only this type of dynamic models.

We consider the following generic representation of the dynamic setup, and the estimation process is considered in that context:

$$ y_{t} = \Gamma y_{t - 1} + w_{t} ,\quad \left( {\text{State equation}} \right) $$

(A.1)

$$ z_{t} = A_{t} y_{t} + v_{t} ,\quad \left( {\text{Measurement equation}} \right) $$

(A.2)

In this dynamic model, $y_{t}$ is a $p \times 1$ vector of unobserved state variables, $\Gamma$ is the $p \times p$ state transition matrix governing the evolution of the state vector. $w_{t}$ is the $p \times 1$ vector of independently and identically distributed zero-mean normal vector with covariance matrix $Q$. The state process is assumed to have started with the initial value given by the vector,$y_{0}$, taken from normally distributed variables with mean vector $\mu_{0}$ and the $p \times p$ covariance matrix,$\Sigma_{0}$.

The state vector itself is not observed but some transformation of these is observed but in a linearly added noisy environment. In this sense, the $q \times 1$ vector $z_{t}$ is observed through the $q \times p$ measurement matrix $A_{t}$ together with the $q \times 1$ Gaussian white noise $v_{t}$, with the covariance matrix,$R$. We also assume that the two noise sources in the state and the measurement equations are uncorrelated.

The next step is to make use of the Gaussian assumptions and produce estimates of the underlying unobserved state vector given the measurements up to a particular point in time. In other words, we would like to find out, $E\left( {y_{t} |\left\{ {z_{t - 1} ,z_{t - 2} \cdots z_{1} } \right\}} \right)$ and the covariance matrix, $P_{t|t - 1} = E\left[ {\left( {y_{t} - y_{t|t - 1} } \right)\left( {y_{t} - y_{t|t - 1} } \right)^{\prime } } \right]$. This is achieved by using Kalman filter and the basic system of equations is described below.

Given the initial conditions $y_{0|0} = \mu_{0} ,{\text{ and P}}_{{0|0}} = \Sigma_{0}$, for observations made at time 1, 2, 3…T,

$$ y_{t|t - 1} = \Gamma y_{t - 1|t - 1} , $$

(A.3)

$$ P_{t|t - 1} = \Gamma P_{t - 1|t - 1} \Gamma^{\prime} + Q, $$

(A.4)

$$ y_{t|t} = y_{t|t - 1} + K_{t} \left( {z_{t} - A{}_{t}z_{t|t - 1} } \right),\quad {\text{where the Kalman gain matrix}} $$

(A.5)

$$ K_{t} = P_{t|t - 1} A^{\prime}_{t} \left[ {A_{t} P_{t|t - 1} A^{\prime} + R} \right]^{ - 1} , $$

(A.6)

and the covariance matrix $P_{t|t}$ after the tth measurement has been made is,

$$ P_{t|t} = \left[ {{\rm I} - K_{t} A_{t} } \right]P_{t|t - 1} $$

(A.7)

Equation (A.3) forecasts the state vector for the next period given the current state vector. Using this one step ahead forecast of the state vector it is possible to define the innovation vector as,

$$ \nu_{t} = z_{t} - A_{t} y_{t|t - 1} $$

(A.8)

and its covariance as,

$$ \Sigma_{t} = A_{t} P_{t|t - 1} A^{\prime}_{t} + R $$

(A.9)

Since in finance and economic applications all the observations are available, it is possible to improve the estimates of state vector based upon the whole sample. This is referred to as Kalman smoother and it starts with initial conditions at the last measurement point i.e. $y_{T|T} {\text{ and P}}_{{\text{T|T}}}$. The following set of equations describes the smoother algorithm:

$$ y_{t - 1|T} = y_{t - 1|t - 1} + J_{t - 1} \left( {y_{t|T} - y_{t|t - 1} } \right), $$

(A.10)

$$ P_{t - 1|T} = P_{t - 1|t - 1} + J_{t - 1} \left( {P_{t|T} - P_{t|t - 1} } \right)J^{\prime}_{t - 1} ,\quad where $$

(A.11)

$$ J_{t - 1} = P_{t - 1|t - 1} \Gamma^{\prime}\left[ {P_{t|t - 1} } \right]^{ - 1} $$

(A.12)

It should be clear from the above that to implement the smoothing algorithm the quantities $y_{t|t} {\text{ and P}}_{{\text{t|t}}}$ generated during the filter pass must be stored.

With reference to the dynamic model, it is obvious that the parameters of interest are embedded in the matrices ${\text{A}}_{t} {\text{, Q, and R}}$. The description of the above filtering and the smoothing algorithms assumes that these parameters are known. In fact, we want to determine these parameters and this is achieved by maximizing the innovation form of the likelihood function. The one step ahead innovation and its covariance matrix are defined by the Eqs. (8) and (9) and since these are assumed to be independent and conditionally Gaussian, the log likelihood function (without the constant term) is given by,

$$ \log \left( L \right) = - \sum\limits_{t = 1}^{T} {\log \left| {\Sigma_{t} \left( \Theta \right)} \right|} - \sum\limits_{t = 1}^{T} {\nu^{\prime}_{t} \left( \Theta \right)} \Sigma_{t}^{ - 1} \left( \Theta \right)\nu_{t} \left( \Theta \right) $$

(A.13)

In this expression $\Theta$ is specifically used to emphasize the dependence of the log likelihood function on the parameters of the model. Once the function is maximized with respect to the parameters of the model, the next step of smoothing can start using those estimated parameters.

Maximization of the function in (A.13) may be achieved using one of two approaches. The first one depends on algorithms like Newton–Raphson and the second one is known as the EM (Expectation Maximization) algorithm. In this paper we employ the Newton–Raphson technique to achieve our objective and since the likelihood function is reasonably well behaved, maximization is achieved quite quickly. In some modelling situations it may not be so straightforward. EM algorithm has been reported to be quite stable in the presence of bad starting values, although it may take longer to converge. Some researchers report that when good starting values are hard to obtain, a combination of the two approaches may be useful. In that situation it is preferable to employ EM algorithm first in order to obtain an intermediate estimate and then switch to Newton–Raphson method. Interested readers may refer to Shumway and Stoffer (2000, p. 323).