Gasoline Price as a Leading Indicator of Mortgage Rate
This paper tests the statistical significance of using residential gasoline price as a leading indicator for predicting Mortgage Interest Rate in the US beyond what the long-term bond yield signals, especially the rate on 10-year U.S. government bond. In particular, time-varying betas are estimated by using Error Correction Model and the Kalman filter technique which is the special case of the general state-space model in order to investigate the significance of the temporal relations between the mortgage interest rate and the gasoline price. Empirical results show that the predicting power in the mortgage interest rate movement can be significantly increased by adding the short-run movement of gasoline price, whereas the role of 10-year bond in predicting mortgage interest rate is gradually decreasing for the last decade. However, the 10-year bond rate is still significant effect on predicting the long-run trend in mortgage interest rate but the long-run relationship between gasoline price and mortgage interest rate is virtually weak.
Keywords: Mortgage Interest Rate, General-to-Specific Model selection, Cointegration, Kalman Filter, Error Correction Model, Gasoline Price, Fisher effect
Good predictions of mortgage interest rate are important to both the lenders and to the borrowers, and also to the policy makers. Mortgage lenders, for example, need to gauge the future direction of mortgage interest rates when considering how much they guarantee the rates for a specific period and how to maximize the market revenue. With respect to closing a mortgage loan, the borrowers’ timing to lock-in
II. Literature Review
A vast literature examines the size of nominal interest rates to changes in expected inflation; broadly known as the Fisher Effect (Grebler, 1983; Johnson, 2005). In particular, most of these studies point out that there exists a long-run comovement between nominal interest rates and inflation rates by using cointegration methods. Gambacorta (2004) shows that the reason for the fall in bank interest rates is due to the fall in inflation via strong monetary policy and that the interest rate on bank loans depends positively on inflation and real GDP. Schwab (1982) and Cohn and Lessard (1976) examine the increasing effects in the mortgage interest rates with the anticipated inflation to compensate the decrease in value of the mortgage payments in real terms. Sun (1992) also points out that the rise in expected inflation raises the nominal interest rate.
When the interest is focused on high-frequency measures of mortgage interest rate, two premises can be considered. In particular, when the prediction of mortgage interest rate for the daily up and down movement is mattered, the typical inflation data measured on a monthly basis is largely uninformative in choosing a lock-in time in loan application. The first premise is the positive relationship between expected inflation and gasoline price, and the second is the positive relationship between nominal interest rates and long-term mortgage interest rates.
For the first premise, Hamilton (2003) and Hamilton and Herrera (2004) show that the main driving force of inflationary pressure in the US economy is the oil price shocks. Castillo, Montoro, and Tuesta (2005) empirically examine the positive relation between the volatility of oil price and inflation premium. Instead of using the monetary policy as the leading indicator of the inflation premium, they use the shocks in oil price to explain the leading cause of inflation premium. Hooker (2002) also tests that oil price has substantial direct effect on the core inflation before 1981 and little effect after that period due to the shift in the monetary regime. These empirical studies show that there is clear comovement between gasoline price and inflation.
For the second premise, many studies are focused on the valuation of mortgage-backed security model. Beginning with Brennan and Schwarts (1979, 1982) on the pricing of bonds, many other studies have advanced models for the pricing of interest-rate sensitive mortgage-backed securities (Brennan and Schwarts, 1985; Green and Shoven, 1986; Riedy, 1983; Schwarts and Torous, 1992). Marthe and Shawky (2003) point out that mortgage rates, as measured by the conventional mortgage rates, are shown to closely follow the long-term interest rates as represented by the ten year Government bond rate. In addition, they examine that changes in the short-term had little or no direct effect on mortgage rates.
Since investors’ willingness to invest in these mortgage-backed securities depends on the interest they yield and the risk premium, the investment in these mortgage bonds go up as the rate increases. During these periods of inflated bond rate, banks can pool a large mortgage fund to finance mortgages, but a high yielding bond lowers the difference between the mortgage interest rate that banks will receive and the bond rate that bank has to pay. Therefore, with a high yielding bond rate, mortgage institutions usually raise the mortgage interest rate to compensate the loss due to lessening gap between mortgage rate and bond rate. For our analytical approach, we consider ten-year bond rate to represent the mortgage backed security rate as the rate usually varies with the ten-year bond rate in the same direction.
II. The Empirical Framework and Data
The basic empirical tests to assess whether gasoline prices lead mortgage interest rate in the very short term are composed of two parts: A) error correction model and B) estimating time-varying coefficients investigating the conditional relationship of time varying parameters between mortgage interest rate and the regressors, residential gasoline price and ten-year bond rate.
A. Error Correction Model
Following the literature that examines the determinants of mortgage interest rate, we postulate a long-run relationship between residential mortgage interest rate and two generally important explanatory factors: the ten-year Government bond rate and the residential gasoline price as follows:
where M is mortgage interest rate, G is the residential gasoline price and B is the ten-year Government bond rate and t denote week identifiers, respectively. The test for a long-run relationship described by equation (1) requires a test for the stationarity of the series. If the series is integrated of the same order, a cointegrating vector might be then found such that a linear combination of the on-stationary variable obtained with that vector is, itself, stationary.
The equation (1) can be written as an autoregressive-distributed lag model by considering long memory process in the market. We propose maximum lag lengths of 1 for the dependent variable and 2 for each exogenous variable. The model (1) is equivalently written by using an error correction model (ECM) as:
where ∆ denotes the difference operator. That means, the equation (2) is the same with the following ECM parameterization conditional on stationary assumption in the differenced data.
The Equation (2’) is a typical format of ECM. Hence, we can interpret β1+β2 and β3 +β4 as the short-run effect of gasoline price and ten year bond on mortgage interest rate, β6/β5 and β7/β5 as the long-run effects. In addition, the coefficient β5 can be interpreted as the speed of adjustment factor for the residual terms in the stationary long-run cointegration process between the dependent variable and the explanatory variables which is depicted in equation (1).
We can incorporate our basic ECM into a more general form of ECM specification with the specific structural break point of mortgage rate process. Thus, we write the most general model as:
where D is the dummy variable to capture the structural break in the mortgage interest rate. Since the estimation of equations (2) and (3) are rationalized only by establishing the existence of the long-run relationship given by equation (1), establishing cointegration between non-stationary processes is a necessary condition for the estimation of an error correction model such as equation (3) (Engle and Granger, 1991). Without cointegration, the statistical properties in equation (3) may be spurious. Hence, we test for cointegration by using the procedure developed by Kanioura & Turner (2003) and proposed by Engle and Granger (1987) and Kremers et al. (1992). These authors show that the cointegration testing based upon a conventional F-test for the joint significance of the levels terms is advantageous because its distribution does not depend on the specific parameters of the problem being considered. We use OLS to estimate equation (3) to find the significance of the explanatory variables.
The final forms of model specifications are obtained from ‘general-to-specific’ specification search which has been popularized particularly by David Hendry (Hendry, 1995; Mizon, 1995; Hendry, 1993; Hendry and Mizon, 1990). Hendry and Krolzig (2001) has recommended the use of multiple search paths in the process of moving from a Generalised Unrestricted Model (GUM) to a parsimonious specification. The reason for this recommendation is to avoid the risk of deleting an important variable which should ideally be retained in the final specification along any single search path and to minimize the risk of retaining as proxies for the missing variable with the result that the final model is overparameterised. Therefore, our final form of the general ECM equation (3) is determined by parsimony, satisfactory performance against diagnostic tests incorporated with Schwarz criterion, evidence of cointegration and the implied long-run relation of equation (1).
B. Time-Varying Parameters (TVP) by using Kalman Filtering
The Kalman filter is a special case of the general state-space model that is composed of the measurement equation and transition equation. The transition equation has the form of a first-order difference equation in the state vector. Consider the following measurement equation of state-space model.
where ∆Mt is the first difference of mortgage interest rate observed at time t and time varying parameter vectors, and the ts are unobserved state variables that explain the variation of the change in the mortgage interest rate. The number of unobserved state variables are equal to the number of explanatory variables in ECM equation (3).
The transition equations that describe the evolution of the time-varying state vector assume the following simple form of a first-order difference equation in the state vector.
The Kalman filter estimates the unobserved state variables t through recursive procedure using maximum likelihood estimators (MLE) method based on the prediction and updating. The maximized log likelihood function is represented by
where t|t-1 is the prediction error and ft|t-1 is the conditional variance of the prediction error. The prediction
C. The Data
In order to determine which way mortgage interest rates are headed when the lock-in time is mattered, high frequency data should be considered. We consider the weekly data from the first week of January 1993 to the first week of September 2005. The mortgage rate is the weekly data for the US national average of 30-year residential fixed rate from mortgage-x (an independent research company on mortgage and housing). The gasoline price data is
IV. Empirical Results
A. ECM Model
The estimates of the equation (3) obtained from a general-to-specific specification search based upon all the diagnostic tests are reported in table 1. The estimated equation passes the test for functional form misspecification, non-normality and cointegration test but not for the first-order serial correlation, and heteroskedasticity of residuals due to high frequency data property. Therefore, following Newey and West (1987), we estimated general form of Heteroskedasticity and Autocorrelation Consistent covariance estimator that is consistent in the presence of both heteroskedasticity and autocorrelation of unknown form. The column 3 in the table 1 shows the Newey-West t-values. Each test statistic is evaluated at the 5% significance level. All the variables in the final equations are tested whether the processes of mortgage interest rate, gasoline price and government 10-year real bond rate are I(0) or I(1). The stationarity of all the variables are verified by using the following tests: Augmented Dickey-Fuller (ADF), Phillips-Perron (PP), Kwiatkowshi, Phillips, Schmidt and Shin (KPSS) and Ng-Perron (NP). All the test results show that the mortgage interest rate, gasoline price and 10-year bond rate are all I(1). In particular, the non-stationarity result in mortgage interest rate is consistent with other empirical studies such as Ledesma (1994), Munro and Tu (1996), and Kenny (1999) and Mcgibany and Nourzad (2004). This suggest that inferences drawn from emprirical studies done using the level form of the variables may not provide an accurate representation of the relation among the variables and verify using ECM.
The test statistics from the residuals of the cointegration results are crucial to verifying the final specification model. The findings show that the null hypothesis of non-cointegration can be rejected at 5% or 10% level of significance. In addition, the typical speed of adjustment factor is significantly less than zero, -0.116. This means that the null hypothesis of non-cointegration can be rejected using the critical values from Banerjeee et al. (1994). These results suggest that there is strong evidence for the existence of a long-run cointegration relationship among the variables. In particular, the long-run cointegrating relationship includes the growth rate of gasoline price, and the current and lagged 10-year bond rate. The DW statistics is higher than the R2, which suggests the existence of a cointegration relationship (Sargan and Bhargava, 1983). As Sargan and Bhargava (1983) point out, DW will approach zero as the sample size increase if the residuals are non-stationary. That means that the DW statistics from the cointergrating regression can be used as an alternative cointegrating regression test.
Table 1: OLS Estimation of ECM (dependent variable: ∆Mt)
Note: t-statistics in bracket. ∆ indicates first difference. REST is Regression Specification Error Test proposed by Ramsey (1969). NRM is the Jarque-Bera statistic for testing normality. LM is a Lagrange multiplier (LM) test for autoregressive conditional heteroskedasticity (ARCH) in the residuals (Engle 1982). W is a test for heteroskedasticity in the residuals from a least squares regression (White, 1980). ECM is the test statistics for Engel-Granger cointegration test. The critical values are -5.75 at 1%, -4.53 at 5% and -.3.99 at 10%.
In order to see the stability of the effect of gasoline price and 10-year bond yield, the cumulative sum of squares (CUSUMSQ) test of Brown et al (1975) was used. Indeed, the test results confirm the general result that the estimated coefficients are not stable over time. Figure 1 shows the result of CUSUMSQ test for the generic ECM without having the dummy variable. In the figure, it can be seen that the plot of the recursive residuals first breaches the 5% boundary in approximately the mid 1995 and stay outside until June of 2003, indicating a shift in regime around June of 2003. This result verifies the use of dummy variable in the equation (3) and TVP approach in analyzing the dynamic effects in the coefficients.
Figure1: The CUSUMSQ Test for ECM Coefficients
The classical relation between 10-year bond rate and the mortgage interest rate are significant and positive. For the short-run dynamic effect of gasoline price, it is positive and significant even though the size is much smaller than the coefficients on the 10-year bond rate. This shows that short-run variations in the gasoline price can be used to predict the same instantaneous changes in the mortgage interest rate. The long-run effect of gasoline price on mortgage interest rate is significant at 10% but not at 5%. In addition, the size of the long-run cointegrated coefficient of gasoline price is much smaller than the size of the coefficient on 10-year bond. That means, the conventional mortgage rates are shown to closely follow the long-term interest rates as represented by the 10-year government bond rate but changes in the gasoline price have little effect in the long term on mortgage interest rate. However, when the lock-in time in loan application is mattered, the movement of gasoline price has significant effect on predicting up or down direction in the mortgage interest rate.
B. TVP Model
Even though most of the estimated coefficients are significant and in the right direction, the ECM estimates are based on a fixed-coefficient model and thus provide only averages for the variable coefficients for the whole sample period. In order to see the dynamic effects of the explanatory variables, it is therefore desirable to examine how the estimated coefficients vary over time as the mortgage interest rage changes over the sample period.
Figure 2 presents plots of the time-varying regression coefficients generated by using the Kalman filter method. Figure 2 shows the short-run dynamic effects of gasoline price and 10-year government bond rate on the mortgage interest rate. As can be seen in these figures, the time-varying coefficients on the 10-year government bond have positive sign over the most of sample period. For the time-varying coefficient on 10-year bond rate, it starts from a low level of 0.28 and significantly decreases to 0.067 until the beginning of 2003 and stays more or less at the same level. However, the time-varying coefficient on gasoline price start form a negative values around the mid 1995 and stays more or less at the level of 0.04 through 2000 and then drop to 0.02 in January of 2002 and then consistently rises over 0.05. These findings explains that the predicting power in the mortgage interest rate can be significantly increased by adding the short-run movement of gasoline price since 2001, whereas the role of 10-year bond in predicting mortgage interest rate is gradually decreasing since 1994.
Figure 2: The TVP Coefficients of Short-run Effects on Mortgage Interest Rate
Figure 3: The TVP Coefficients of Long-run Effects on Mortgage Interest Rate
Figure 3 shows the long-run cointegrated relationship between mortgage interest rate and the regressors, gasoline price and 10-year government bond rate. These figures show that the 10-year bond rate is still significant effect on predicting the long-run trend in mortgage interest rate but the gasoline price is unclear in the long-run relationship. The long-run time varying movement of the coefficient on gasoline price is mostly negative until 2000 and then stays more or less at the level of 0.05 whereas the TVP on 10-year bond rate is around 0.76 until 1998 and drop 0.52 in 2000 and it increases to 0.65. These two plots suggest that the long-run movement of mortgage interest rate is driven mainly by the long-run trend of long-term interest rate. These results are consistent with the results in ECM estimation.
This paper examines the dynamic relationships among conventional mortgage rates, the residential gasoline price, and the long-term interest rates by using Error Correction Model and Time Varying Parameter of Kalman Filtering. This paper explores whether the variations in residential gasoline price add further information about future direction of mortgage interest rate beyond what the long-term bond yield signals, especially the rate on 10-year U.S. government bond. In particular, this paper is focused on whether gasoline price can be used as a leading indicator of mortgage rate.
Several important conclusions emerge. The most important finding of the study is that the predicting power in the mortgage interest rate movement can be significantly increased by adding the short-run movement of gasoline price, whereas the role of 10-year bond in predicting mortgage interest rate is gradually decreasing since 1994. This result suggests that the lenders, the barrowers, and the policy makers can easily check the street price of residential gasoline price in order to see which way mortgage interest rates are headed in the very short term.
Another interesting finding is that that the 10-year bond rate is still significant effect on predicting the long-run trend in mortgage interest rate but the gasoline price is unclear in the long-run relationship. The TVP model of Kalman filtering confirms that the dependence of the mortgage interest rates on changes in the long-term rates is significant in the long-run but it is gradually getting smaller in the short-run, whereas the dependence of the gasoline price on changes in the long-term rates has been increasing since 2001 but it has very small explanatory power in the long-run.
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