Factors associated in housing market dynamics: an exploratory longitudinal analysis.

Choudhury, Askar

INTRODUCTION

In this study, I propose a hypothetical model to examine the association of various determinants of housing starts as a measure of core housing market to study the housing market dynamics. Although, macroeconomic factors are commonly viewed as important causes of housing market movements, other factors may also be important driver of the housing market. Incorporation of demographic and macroeconomic factors may enhance housing market models' performance. However, long-term momentum may be additionally associated with endogenous factors, such as, number of houses sold and number of houses for sale. This inter-dependent market activity is recursive in nature and creates domino effect (Choudhury & Campbell, 2004) to push the market further upward/downward depending on the market condition. Evidently, there are various interactions between these factors, which may or may not be observable. Some of these unobservable factors are embedded in the number of houses sold and number of houses for sale; whose developments may be shaped by economic, demographic, and other factors. Capturing these unobserved components effect (indirectly) is the primary goal of this study. In that regard, I propose a multivariate cross-correlation time-series approach. Understanding this complex recursive phenomenon between these factors would assist housing lenders in assessing the risk of default and investment portfolio managers in assessing the direction of market movements. Once the magnitude of the effect of these key determinants inter-dependent association is well understood, government policy makers could induce the market stability by adjusting the market environment accordingly.

To my knowledge, this is the first study to report differential effects of number of houses sold and number of houses for sale on the housing starts. In particular, using cross-correlation analysis, I find the relationship between housing starts and the number of houses sold is positively correlated (as number of houses sold increase, housing starts increase) at least for two years. Moreover, the data show a strong inverse lead-lag relationship between housing starts and the numbers of houses for sale after several months lag (as number of houses for sale increase, housing starts decrease). This exhibits long-term statistical dependence; however, I find the magnitude and the nature of the dependency differs between number of houses sold and number of houses for sale. These cross-correlations are not widely known and suggest an additional link between housing starts and unobservable factors.

Cross-correlation analysis reveals that the association between housing starts and the number of houses sold are strongly positive and immediate, and it continues to persist for over two years. In contrast, the association between housing starts and the number of houses for sale are initially weakly positive, but after six months of delay it becomes negative and the association continues to grow stronger negatively for over two years. In addition to cross-correlation analysis, I perform time-series regression analysis (see, Choudhury, Hubata, & St. Louis, 1999 for more on time-series regression) to identify the influential lag effect on the housing starts. I find statistically significant but inverse association between housing starts with number of houses sold and number of houses for sale. These results suggest the impact of number of houses sold on the housing starts is different both in direction and also in magnitude.

Thus, the objective of this paper is to examine the direction and magnitude of lead-lag association between housing starts with number of houses sold and number of houses for sale. To my knowledge, no research has been done to analyze and test the differential lead-lag effect of number of houses sold and number of houses for sale on the housing starts, which is the core of housing market dynamics. Therefore, this research primarily focuses on identifying the length of lead-lag effect of these factors on the housing starts and also the direction and magnitude of these effects.

LITERATURE REVIEW

The basic dichotomy of housing starts (specifically single-family starts) can be characterized into speculative housing starts for investment purposes (by investors or builders) and owner initiated custom-built housing starts. Research suggests that, the volatility (or instability) in the housing market is largely attributable to the speculative portion of the housing starts. This segment of the housing market creates its own dynamics with relations to number of houses for sale and therefore with number of houses sold. Thus creating a lead-lag relationship among these factors that persists over several months. These considerations posit lead-lag relations between number of houses sold and number of houses for sale with the housing starts that facilitate a partial explanation of housing market's rapid movements. In a recent report, Congressional Budget Office (CBO) stated that, "Starts of new housing units peaked at an annual rate of just over 2.1 million in the first quarter of 2006, buoyed by low mortgage interest rates, expectations of continued rapid increases in home prices, and lax lending standards. By the second quarter of 2008, lower expectations of home price increases and tighter lending conditions had combined with a glut of vacant units to cut housing starts by more than half, to an annual rate of barely 1.0 million."

Housing market plays a significant role as leading indicator of the economy, and therefore understanding the market dynamics cannot be overemphasized, especially in light of the recent housing market turmoil and its effect on the economy as a whole. Since, the movements in the housing market will likely continue to play an important role in the business and economy (Gupta & Das, 2009; Bernanke and Gertler, 1995), understanding the market mechanism, specifically the lead-lag relationship between factors can offer policy makers a notion about the direction of the overall market trajectory in advance, and thus, provides a better control for designing appropriate policies for market stabilization.

As a result of such importance of the housing market on the economy, a large number of studies on the housing market have been undertaken recently. In recent years, researchers have devoted much of their effort to identify factors that determine the housing market mechanism (Sander & Testa 2009; Lyytikainen, 2009; Fratantoni & Schuh, 2003; Taylor, 2007; Bradley, Gabriel, & Wohar, 1995; Vargas-Silva, 2008). Many factors have been cited (Ewing & Wang, 2005; Baffoe-Bonnie, 1998; Huang, 1973; Thom, 1985) as sources of housing market dynamics; among these, housing price (Rapach & Strauss, 2009) and housing starts (Lyytikainen, 2009; Ewing & Wang, 2005; Puri & Lierop, 1988; Huang, 1973) play a very important role. These studies have been primarily designed to examine particular aspects of these markets, such as the relationship between residential construction and credit accessibility (Taylor, 2007; Guttentag, 1961; Alberts, 1962; Thom, 1985; Mayer & Somerville, 1996), magnitude of the demand elasticity with respect to price and income (Sander & Testa 2009; Mankiw & Weil, 1989; Meen, 2000; Reid, 1958; Lee, 1964; Mulligan & Threinen, 2008;), and the determinants of housing starts (Rapach & Strauss, 2009; Addison-Smyth, McQuinn, & O'Reilly, 2008; Dipasquale, 1999; Kearl, 1979; Maisel, 1963).

Overall, empirical evidence suggest a contemporaneous positive association between number of houses sold and number of houses for sale with housing starts. However, time-series investigations have delayed autocorrelation effect. Therefore, the purpose of this paper is to understand the cross-correlation dynamics of housing market with particular emphasis placed upon the role of housing starts. Specifically, using the research design discussed in the following section, the present study attempts to isolate particular lead-lag association between number of houses sold and number of houses for sale with housing starts.

DATA AND RESEARCH METHODOLOGY

The sample period is a time series of monthly data beginning January 1991 and ending April 2009. Limiting the sample period to these years, avoids certain shortcomings of missing data in some factors. Data are collected from the US Census Bureau and Federal Reserve Board. I have selected the new privately owned housing units start (Housing Starts) as my measure of housing market dynamics. Housing starts is most widely used factor in understanding the dynamics of housing market (Ewing & Wang, 2005; Fullerton, Laaksonen, & West, 2001; Mayer & Somerville, 1996; Vargas-Silva, 2008). Home builders would respond to the market demand when constructing new homes and the decision for new starts may depend on the accelerated /decelerated rate the number of houses are being sold and/or increased/decreased number of houses for sale on the market. Consequently, these decisions take time to be implemented and as a result housing starts adjust to these changes after several months delay. Thus, the objective of this paper is to understand the housing market dynamics and their delayed response to housing starts. In addition to these factors, model also incorporated control variables, such as, civilian employment to population ratio and mortgage rate. Mortgage rate is found to be most effective at lag 6 (see, Table 3).

Table 1 shows the distributions of housing starts, houses sold, houses for sale, civilian employment to population ratio, and mortgage rate for the sample period. As observed in Table 1, average number of houses sold exceeded the average number houses for sale approximately by 3:1 margin. Also, the number of houses sold per month shows more variance than the number of houses for sale. Table 1 also presents the summary statistics for mortgage rate and civilian employment to population ratio.

I hypothesize that the number of houses sold and the numbers of houses for sale are inversely associated with housing starts. To test my hypothesis I perform two separate analyses. First, I use the cross-correlation analysis to examine the direction of the association and whether the number of houses sold and/or the number of houses for sale exhibit any long memory, a term refers to long-term statistical dependence in time series data. Second, I use time-series regression to examine the magnitude and significance of housing starts using other factors over time and to observe any acceleration /deceleration of the momentum of the process. Specifically, I regress the housing starts on the number of houses sold (House Sold) and the number of houses for sale (House for Sale), after controlling for mortgage rate and civilian employment to population ratio. Increase in civilian employment to population ratio indicates increasing capacity of possible homeownership. On the other hand, increase in mortgage rate indicates decreasing capacity of possible homeownership.

In an effort to better disentangle the effects of housing starts momentum from expanding or contracting housing market activity, regression model includes these control variables measuring the market capacities. Additionally, Durbin-Watson statistic of ordinary least squares (OLS) estimates indicated the presence of positive autocorrelation. One major consequence of autocorrelated errors (or residuals) when applying ordinary least squares is the formula variance [[[sigma].sup.2] [(X' X).sup.-1]] of the OLS estimator is seriously underestimated (see Choudhury, 1994), which affects statistical inference. Where X represents the matrix of independent variables and o 2 is the error variance.

Durbin-Watson statistic is not valid for error processes other than the first order (see Harvey, 1981; pp. 209-210) process. Therefore, I evaluated the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the OLS regression residuals using SAS procedure PROC ARIMA (see SAS/ETS User's Guide, 1993). This allowed the observance of the degree of autocorrelation and the identification of the order of the residuals model that sufficiently described the autocorrelation. After evaluating the ACF and PACF, the residuals model is identified as second order autoregressive model: (1 - [[phi].sub.1] B - [[phi].sub.2] [B.sup.2])[v.sub.t] = [[epsilon].sub.t] (see Box, Jenkins, & Reinsel, 1994). The final specification of the regression model takes the following form:

[HStart.sub.t], = [[beta].sub.0] + [[beta].sub.1] [CEPR.sub.t] + [[beta].sub.2] [MTG.sub.t-6] + [[beta].sub.3] [HS.sub.t], + [[beta].sub.4] [HS.sub.t-3] + [[beta].sub.5] [HS.sub.t-6] + [[beta].sub.6] [HFS.sub.t-24] + [v.sub.t] and [v.sub.t] = [[phi].sub.1][v.sub.1-1] + [[phi].sub.2] [v.sub.t-2] + [[epsilon].sub.t]

Where: HStart = number of housing starts, CEPR= civilian employment to population ratio, MTG= mortgage rate, HS= number of houses sold, HFS= number of houses for sale, and (t-k) is for k months lag or delay.

Maximum likelihood estimation method is used instead of two step generalized least squares to estimate the regression parameters in the regression model. Maximum likelihood estimation is preferable over two step generalized least squares, because of its capability to estimate both regression and autoregressive parameters simultaneously. Moreover, maximum likelihood estimation accounts for the determinant of the variance-covariance matrix in its objective function (likelihood function). Further discussion on different estimation methods and the likelihood functions can be found in Choudhury, Hubata & St. Louis (1999); also SAS/ETS User's Guide, 1993 for the expression of the likelihood functions. Likelihood function of the regression model with autocorrelated errors can be expressed as follows:

L([beta], [theta], [[sigma].sup.2] = - n/2 ln ([[sigma].sup.2])) - 1/2 ln [absolute value of [OMEGA]] - (Y - X[beta])' [[OMEGA].sup.-1] (Y - X[beta])/2[[sigma].sup.2]

where,

Y- vector of response variable (housing starts),

X--matrix of independent variables,

[beta]--vector of regression parameters,

[theta]--vector of autoregressive parameters,

[[??].sup.2]--error variance,

[OMEGA]--variance-covariance matrix of autocorrelated errors.

EMPIRICAL ANALYSIS

I report the results of statistical analysis investigating the association between housing starts, number of houses sold, and number of houses for sale. Table 2 presents' lead-lag correlations along with their p-values (in parentheses) for housing starts with number of houses sold and number of houses for sale up to 24 months lag. Strong positive correlations are observed with housing starts and the number of houses sold. Even though the association remains statistically significant up to 24 months lag, the strength of the association diminishes slowly indicating the impact on housing starts is more pronounced during the recent months than past. In contrast, correlations between housing starts and the number of houses for sale is negative but not immediate, the impact is delayed. Thus, the number of houses for sale show a weak positive correlation initially; however, they exhibit long memory in the opposite direction (after six months delay) and the strength of the relationship continues to increase negatively as the delay (or lag) gets longer. The concept of long memory in a time series is used to indicate statistical dependence in which the autocorrelation function decays at a much slower rate than in the case of short-term statistical dependence. Long-term dependence has only begun to be addressed recently in macroeconomic and financial time series data (Abderrezak, 1998). The negative impacts of number of houses for sale on housing starts become statistically significant after nine months and remain strong over two years. Delayed negative impact is consistent with the idea that more houses for sale in the market increases the supply of houses and consequently impacts the number of new houses to be built. This result is consistent with other research findings in that it suggests protracted upward (or downward) spiral (Taylor, 2007) momentum of the market mechanism known as domino effect (Choudhury & Campbell, 2004).

Regression results reported in Table 3 provides confirming evidence of the contrasting effect of number of houses sold and number of houses for sale on the housing starts. Civilian employment to population ratio is positively associated with housing starts; however, mortgage rate (delayed by six months) is negatively associated with the housing starts. Similar results are also reported by other researchers (Mayer & Somerville, 1996). I applied forward, backward, and mixed stepwise methods to select the regression model through the R-squared statistics and significance level as a criterion to add variables into the model or delete variables from the model. All three types of stepwise methods yielded the same result. Moreover, the model resulting from stepwise selection provided the same conclusion that number of houses sold, number of houses for sale, civilian employment to population ratio, and mortgage rate are significant factors in impacting the likelihood of housing starts. Number of houses sold and civilian employment to population ratio have direct impact on the housing starts, as indicated by the positive coefficients that resulted in increasing housing starts. More specifically, one can assert that if the civilian employment to population ratio increases by one percent, housing increases by approximately 35,755 new starts. Contrary to that, number of houses for sale and mortgage rate has opposite (or negative) impact on the housing starts, as indicated by the negative coefficients that resulted in decreasing housing starts. These results suggest if the mortgage rate increases by one percent, new starts on housing decreases by approximately 40,067. After being adjusted for autocorrelation, the Durbin-Watson test-statistic (DW=2.02) indicates that the errors are not correlated. Also, the R-squared statistic of the model is significantly high at 0.94.

CONCLUSION AND DISCUSSION

This paper makes a number of significant contributions to the literature. It provides additional evidence of differential effect of various factors on housing starts. In addition, it also provides evidence suggesting number of houses sold and number of houses for sale display long memory. However, associations between number of houses sold and the numbers of houses for sale with housing starts are inversely related. These results while important are not unexpected given the stormy dynamics of the housing market. The unexpected finding is the initial weakly positive association between housing starts and the number of houses for sale. The association becomes negative after few months delay and continues to rise negatively for over two years.

Considering number of houses sold and number of houses for sale separately from other macroeconomic factors illustrates how state policy makers can benefit from using the results of this study. It is also well known that housing starts is considered to be a important leading indicator, as it is included in the Conference Board's leading economic indicators list. Therefore, understanding the mechanism of lead-lag relationship between factors with housing starts will provide an advantageous position to the policy makers to prepare an appropriate policy design for market stabilization.

Thus, these results add another dimension to the debate concerning the effect of observable and unobservable factors on the housing market activity. Additional theory development is needed, particularly with regard to the linkage between observable and unobservable factors. To determine whether the negative association between housing starts and the number of houses for sale is stationary, future research could examine these relations over different periods of time.

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Askar Choudhury, Illinois State University

Table 1: Summary Statistics for the Periods: January 1991-April 2009
(Monthly Data).

Variable Mean Std Dev Minimum Maximum

Housing Starts 1505 336.556 488 2273
Civilian Employment 62.81991 0.986 59.7 64.7
 to Population Ratio
Mortgage Rate 7.11489 1.092 4.81 9.64
House Sold 829.83636 231.578 329 1389
House for Sale 354.23636 84.955 261 570

Table 2: Lead-lag correlations (p-values) between Housing Starts,
Houses Sold, and Houses for Sale.

 Housing Housing
Monthly Lags Starts Monthly Lags Starts

House Sold Lag0 0.94053 House For Sale Lag0 0.25439
 (<.0001) (0.0001)

House Sold Lag1 0.93443 House For Sale Lag1 0.20766
 (<.0001) (0.0021)

House Sold Lag2 0.92492 House For Sale Lag2 0.15902
 (<.0001) (0.0191)

House Sold Lag3 0.91541 House For Sale Lag3 0.11162
 (<.0001) (0.1018)

House Sold Lag4 0.90094 House For Sale Lag4 0.06570
 (<.0001) (0.3377)

House Sold Lag5 0.87385 House For Sale Lag5 0.02249
 (<.0001) (0.7436)

House Sold Lag6 0.85918 House For Sale Lag6 -0.02319
 (<.0001) (0.7364)

House Sold Lag7 0.83502 House For Sale Lag7 -0.06356
 (<.0001) (0.3571)

House Sold Lag8 0.80808 House For Sale Lag8 -0.10209
 (<.0001) (0.1394)

House Sold Lag9 0.78232 House For Sale Lag9 -0.13989
 (<.0001) (0.0429)

House Sold Lag10 0.75284 House For Sale Lag10 -0.17521
 (<.0001) (0.0112)

House Sold Lag11 0.72523 House For Sale Lag11 -0.20740
 (<.0001) (0.0026)

House Sold Lag12 0.68131 House For Sale Lag12 -0.23885
 (<.0001) (0.0005)

House Sold Lag13 0.64600 House For Sale Lag13 -0.26970
 (<.0001) (<.0001)

House Sold Lag14 0.60894 House For Sale Lag14 -0.29794
 (<.0001) (<.0001)

House Sold Lag15 0.57006 House For Sale Lag15 -0.32482
 (<.0001) (<.0001)

House Sold Lag16 0.53227 House For Sale Lag16 -0.35392
 (<.0001) (<.0001)

House Sold Lag17 0.49198 House For Sale Lag17 -0.37938
 (<.0001) (<.0001)

House Sold Lag18 0.45248 House For Sale Lag18 -0.40268
 (<.0001) (<.0001)

House Sold Lag19 0.41575 House For Sale Lag19 -0.42520
 (<.0001) (<.0001)

House Sold Lag20 0.37478 House For Sale Lag20 -0.44250
 (<.0001) (<.0001)

House Sold Lag21 0.33766 House For Sale Lag21 -0.46141
 (<.0001) (<.0001)

House Sold Lag22 0.30124 House For Sale Lag22 -0.47432
 (<.0001) (<.0001)

House Sold Lag23 0.26239 House For Sale Lag23 -0.48687
 (0.0002) (<.0001)

House Sold Lag24 0.22180 House For Sale Lag24 -0.49977
 (0.0018) (<.0001)

Table 3: Regression Results for Housing Starts (Maximum Likelihood
Estimation).

 Maximum
 Likelihood Approx
 Estimates of Pr >
 Parameters [absolute
Independent Variables (corrected for Standard value
(monthly) autocorrelation) Error t Value of t]

Intercept -974.5772 772.5781 -1.26 0.2087
Civilian Employment 35.7554 13.4185 2.66 0.0084
 to Population Ratio
Mortgage Rate LAG6 -40.0669 19.1881 -2.09 0.0381
House Sold 0.5751 0.1007 5.71 <.0001
House Sold LAG3 0.3994 0.1076 3.71 0.0003
House Sold LAG6 0.1861 0.1053 1.77 0.0788
House for Sale LAG24 -1.3172 0.1850 -7.12 <.0001
R-Squared 0.9404
Durbin-Watson 2.0167

Note: The regression residuals model is identified as,
(1 - [[phi].sub.1]B - [[phi].sub.2][B.sup.2])[v.sub.t] =
[[epsilon].sub.t] and the estimated first and second order
autoregressive (AR) parameters from SAS are,
(1 + 0.2504 B + 0.1898 [B.sup.2]) [v.sub.t] = [[epsilon].sub.t]
3.40 *** 2.56 **.

Autoregressive parameter's t-statistics are reported in the
parentheses. They are both significant at the one (***) percent
and five (**) percent level of significance respectively (