NESUG 18 Analysis
Using SAS ® for Cointegration and Error Correction Mechanism Approaches: Estimating a Capital Asset Pricing Model (CAPM) for House Price Index Returns
By Ismail Mohamed and Theresa R. DiVenti ABSTRACT Many researchers erroneously use the framework of linear regression models to analyze time series data when predicting changes over time or when extrapolating from present conditions to future conditions. Caution is needed when interpreting the results of these regression models. Granger and Newbold (1974) discovered the existence of ‘spurious regressions’ that can occur when the variables in a regression are nonstationary. While these regressions appear to look good in terms of having a high R 2 and significant t-statistics, the results are meaningless. Both analysis and modeling of time series data require knowledge about the mathematical model of the process. This paper introduces a methodology that utilizes the power of the SAS DATA STEP, and PROC X12 and REG procedures. The DATA STEP uses the SAS LAG and DIF functions to manipulate the data and create an additional set of variables including Home Price Index Returns (HPI_R1), first differenced, and lagged first differenced. PROC X12 seasonally adjusts the time series. Resulting variables are manipulated further (1) to create additional variables that are tested for stationarity, (2) to develop a cointegration model, and (3) to develop an error correction mechanism modeled to determine the short-run deviations from long-run equilibrium. The relevancy of each variable created in the data step to time series analysis is discussed. Of particular interest is the coefficient of the error correction term that can be modeled in an error correction mechanism to determine the speed at which the series returns to equilibrium. The main finding is that Metropolitan Statistical Areas (MSAs) with very slow short- run acceleration paths to the equilibrium have higher returns and risk associated with house price returns than MSAs with very rapid speed-of-adjustment coefficients.
I INTRODUCTION The purpose of this paper is to develop an approach specifically for one-equation models that can be used in place of the more complex standard SAS program routines PROC ARIMA . The ARIMA methodology developed by Box and Jenkins (1976) has gained enormous popularity in other areas of research attesting to its power and flexibility (Hoff, 1983; Pankratz, 1983). However, because of its power and flexibility, ARIMA is a complex technique; it requires a great deal of experience to accurately use and interpret its results (Bails & Peppers, 1982). The SAS techniques we use in place of the ARIMA for a one-equation model consist primarily of data manipulation and the REG procedure. The SAS programming techniques model Cointegration and an Error Correction Mechanism (ECM).
We demonstrate these techniques using the Capital Asset Pricing Model (CAPM) for MSA and US house price returns, each being a univariate time series. We use the CAPM as the Cointegration Equation in the Engle-Granger two-step procedure (Engle and Granger, 1987). The CAPM was developed by Sharpe (1964), Linter (1965), and Mossin (1966) to estimate the expected returns-beta relationship of individual stocks against a market portfolio. A stock’s beta measures its market or systemic risk. More specifically, a stock beta measures its covariance with the returns of the market portfolio divided by the variance of the market portfolio. In our CAPM model, house price return series for MSAs are used in place of stock returns and the US house price return series is used in place of the market portfolio to estimate MSA specific Betas for returns on a house asset in that location. 1
The MSA-Betas indicate the MSA house investment’s relative systemic risk compared to the US. For MSA-Betas greater than (less than) one, the MSAs expected return is greater (smaller) than the US. A negative MSA-Beta indicates that as the US returns increase, the MSA returns decrease, and vice versa. MSA-Betas provide investors with performance statistics that enable them to assess the risk and return profile associated with the location of their house asset.
1 NESUG 18 Analysis
The house price returns, like most stock returns data, are nonstationary time series. The cointegration between MSA and US house price returns is tested to determine if a long run relationship exists. For those MSAs where the CAPM regression residuals are stationary, the MSA Betas are a consistent estimate of a long-run model.
Once a cointegrating link is determined between a MSA and US house price returns, an Error Correction Model for MSA house price returns is constructed to estimate short-run deviations of house price returns from the long-run equilibrium. Based on the Engle and Granger 2 Representation Theorem, the first difference of the US returns and the residual of the CAPM are regressed on the first differences of the MSA returns. Of particular interest is the coefficient on the CAPM residual, also known as the error correction term. It estimates the quarterly correction of the discrepancy between the actual and long run house price returns.
This paper makes the following contributions. First, the SAS technique utilizes the power of the SAS Data Step and procedures for both data manipulation and analysis. SAS LAG and DIF functions are used to make four adjustments to the dataset and to create a set of new variables including the US and MSA Home Price Index Returns, the US and MSA seasonally adjusted Returns, the US and MSA adjusted Returns’ 1 st , 2 nd , 3 rd , 4 th and 5 th lags as well as the 1 st - differenced of each of the lagged values. Second, the REG procedure is then used (1) to test the US and MSA adjusted house price returns for stationarity, (2) to model the long-run relationship between the US and MSA adjusted Returns, (3) to test the resulting residuals vector for stationarity, and (4) to implement an error correction mechanism to determine the short-run deviations from long-run equilibrium. Details and relevancy of each data adjustment will be fully discussed later in this paper.
II PROCEDURES
This section discusses the four major parts of this analysis. The first part outlines the steps in the data manipulation. The second part discusses the relationship between linear regression and stationarity and the role unit roots play in analyzing time series data. This part is followed by the Cointegration Regression, which shows a long-run link between time series data for MSA and US house price returns. The final section develops an Error Correction Model to estimate the short-run deviation of MSA house price returns from their long-run equilibrium. This section is followed by the Empirical Findings that summarizes our application of these procedures to the Capital Asset Pricing Model and house price returns series. II.A Data Manipulation
There are four adjustments involved in deriving house price returns from the house price indices and computing the level and first differences of the house price returns for use in later sections in the Augmented Dickey Fuller Test, the Cointegration Regression, and the Error Correction Model. These adjustments are discussed below.
The first computes house price returns for the MSAs and the entire US from the Office of Federal Housing Enterprise Oversight (OFHEO) house price indices. Table 1 provides a snapshot of the OFHEO quarterly data for an example MSA, namely, Albany, GA, in column 5 and the US in column 6. OFHEO provides house price indices for 331 MSAs and the entire U.S. 3 The house price index returns are computed based on quarterly data and are calculated as follows:
rhj = (hpi t+ ,1 j − hpi t, j /) hpi t, j (1) where j is the MSA or US index and t is the quarter and year. Each series is made up of 48 returns based on data from first quarter 1988 to fourth quarter 2000.
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Table 1. Snapshot of OFHEO National and Metropolitan House Price Indices Time Series Albany, GA MSA
N_MSA_NAME MSA YEAR QUARTER MSA_HPI USA_HPI Albany, GA 0120 1987 1 85.3 78.612 Albany, GA 0120 1987 2 86.67 80.129 Albany, GA 0120 1987 3 84.91 81.451 Albany, GA 0120 1987 4 82.77 82.215 Albany, GA 0120 1988 1 89.14 83.704 Albany, GA 0120 1988 2 87.91 85.487 Albany, GA 0120 1988 3 87.77 86.397 Albany, GA 0120 1988 4 87.61 87.34 Albany, GA 0120 1989 1 81.48 88.488 Albany, GA 0120 1989 2 86.82 89.658 Albany, GA 0120 1989 3 86.5 91.722 Albany, GA 0120 1989 4 88.27 92.578
The SAS LAG function was used in this adjustment. The function simply looks back in the dataset n number of records and allows you to obtain a previous value for a variable and store it in the current observation. ' n' refers to the number of records back in the data and can be an integer from 1 to 99. Many times the only thing you want to do with a previous value of a variable is to compare it with the current value to compute the difference.
The DIF n function works the same way as LAG n, but rather than simply assigning a value, it assigns the difference between the current value and a previous value of a variable. The statement
At = DIFF (X)
tells SAS that ‘A’ should equal the current value of x minus the value that x had n number of records back in the data. Both LAG and DIFF functions should only be used on the right hand side of assignment statements. They should not be executed conditionally, first execute the LAG/DIFF function and assign the results to a new variable, and then use the new variable for the conditional processing. A sort BY MSA YEAR QUARTER was necessary before this data step.
DATA HPI_INDICES; SET HPI_SORTED; BY MSA YEAR QUARTER; ARRAY MSA_LAG_VEC(4)MSAHPI_LAG1-MSAHPI_LAG4; MSAHPI_LAG1 = LAG (MSA_HPI); MSAHPI_LAG2 = LAG2(MSA_HPI); MSAHPI_LAG3 = LAG3(MSA_HPI); MSAHPI_LAG4 = LAG4(MSA_HPI); ... USHPI_LAG1 = LAG (US_HPI);
IF FIRST.MSA THEN COUNT = 1; DO I = COUNT TO 4; MSA_LAG_VEC (I) = .; END; COUNT+1; MSAHPI_R1 = (MSA_HPI - MSAHPI_LAG1)/MSAHPI_LAG1; USHPI_R1 = (US_HPI - USHPI_LAG1)/USHPI_LAG1;
RUN ;
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Next the resulting MSA house price returns data are seasonally adjusted using the Census X12 series. The SAS X12 procedure is an adaptation of the U.S. Bureau of the Census X-12-ARIMA Seasonal Adjustment program. 4 The purpose of this process is to removing the seasonal effects from a time series data. While the specific functional relationship between these components is either additive or multiplicative, we use an additive functional form .
The SAS X12 procedure creates an output data set containing the adjusted time series and intermediate calculations including a trend cycle (D12), a seasonal factor (D10), and the residual or irregular component (D12). A BY statement is used with PROC X12 to obtain separate analyses on observations by MSA . A sort was necessary by MSA, year, and quarter before the dataset is used in the procedure. Variables specified for the analysis included both the Metropolitan Statistical Areas house price indices quarterly return ( MSAHPI_R1) and US house price indices quarterly return ( USHPI_R1). The X11 statement was issued for invoking seasonal adjustment, options used include MODE= ADD ( mode of the seasonal adjustment). The final seasonally adjusted series (D11) is captured together with the original series (A1) in an output dataset.
PROC X12 DATA=INDX_SORTED DATE=BYQRT SEASONS= 4 SPAN= ('87Q4','03Q4') NOPRINT; VAR MSAHPI_R1; BY MSA; X11 MODE=ADD; OUTPUT OUT=OUT_MSA A1 D11; RUN ;
The third data step involves adjusting the excess returns by the risk free rate. The risk free rate is assumed to be the 10-year treasury rates. Rates for the months March, June, September, and December were used. To make them comparable with the quarterly house price returns, 10-year treasury rates were divided by four such as:
DATA ...; SET ...; ...; USHPI_R10X = USHPI_R1_D11-(TCM10Y/ 4); MSAHPI_R10X = MSAHPI_R1_D11-(TCM10Y/ 4); ... ; RUN ;
In adjustment 4, seven variables were created for each of the house price returns series that will be used in one or more of the three modeling approaches. The SAS LAG and DIF functions were used to create these variables. These variables included: the first lag of the house price return series (R1_1ST_LAG), the first difference of the house price indices return (R1_1ST_DIFF), the first difference of lags 1 –5, namely, R1_DIFF_LAG1, R1_DIFF_LAG2, R1_DIFF_LAG3, R1_DIFF_LAG4, and R1_DIFF_LAG5.
The first difference, first lag of the return series, and the first difference of lag1 - lag5 will be used to carry the ADF test at level with 5 lags, while the first difference of lag1, the second difference the returns series, and the second difference of lag1-lag5 will be used to carry the ADF test at first difference with 5 lags for MSA and US, series. Similarly, the first difference of residual series resulted form modeling the long term relationship between MSA and US return series, together with the first lag, and the first difference of lag1 - lag5 will be used to carry the ADF test at level with 5 lags on residual (cointegration test). Residual series resulted form modeling the long-term relationship between MSA and US return series will be applied to model the short term and error correction relationship.
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DATA ...; SET ...; MSAR_1 st _LAG = LAG1 (MSAHPI_R10X_D11); MSAR_2 nd _LAG = LAG2 (MSAHPI_R10X_D11); MSAR_1st_DIFF = DIF1(MSAHPI_R10X_D11); MSAR_2 ND _DIFF = DIF2(MSAHPI_R10X_D11);
MSAR_1 st _DIFF_1 st _LAG = DIF1 (LAG1 (MSAHPI_R10X_D11)); MSAR_1 st _DIFF_2 nd _LAG = DIF1 (LAG2 (MSAHPI_R10X_D11)); MSAR_1 st _DIFF_3 rd _LAG = DIF1 (LAG3 (MSAHPI_R10X_D11)); MSAR_1 st _DIFF_4 th _LAG = DIF1 (LAG4 (MSAHPI_R10X_D11)); MSAR_1 st _DIFF_5 th _LAG = DIF1 (LAG5 (MSAHPI_R10X_D11)); ...; RUN;
II.B Stationarity and a Unit-Root
The stationarity of a time series has important implications for regression analysis since the classical tests of regression analysis, such as the t-test and F-test, are based on the assumption that time series are stationary. Consequently, the validity of coefficients on explanatory variables is based on stationary series. If, however, a time series process exhibit nonstationarity (i.e., a random walk) standard test statistics are no longer valid and concerns arise over interpreting coefficients that are spurious. 5
To check stationarity, the Augmented Dickey Fuller test (Dickey and Fuller (1981)) is used to determine if the MSAs house price return series are stationary. The Augmented Dickey Fuller (ADF) test is conducted for each of the 331 MSA and the US at the level and first difference. The Augmented Dickey Fuller test is 5
∆r ,tj = κr ,tj −1 + ∑ϖ j,k ∆r ,tj −k + ε k ,t (2) k=1 where the data are said to follow a unit root if κ is not significantly different than 0. In this specification, because the data are quarterly, five lagged differences are included for more explanatory power (for an MSA example, please refer to Tables 2,3, and 4 for ADF test results).
SAS Program: Unit Root Test At Level, And 1 st Difference, Fixed 5 Lags and a Constant
PROC REG DATA = REG_SERIES; *ADF at Level, Fixed 5 Lags and a Constant ; MODEL MSAR_1 st _DIFF= MSAR_1 st _LAGMSAR_1 st _DIFF_1 st _LAG MSAR_1 st _DIFF_2 nd _LAG MSAR_1 st _DIFF_3 rd _LAG MSAR_1 st _DIFF_4 th _LAG MSAR_1 st _DIFF_5 th _LAG;
st *ADF at 1 Difference, Fixed 5 Lags and a Constant; MODEL MSAR_2 nd _DIFF= MSAR_1 st _Diff_1 st _LAG MSAR_2 nd _DIFF_1 st _LAG MSAR_2 nd _DIFF_2 nd_LAG MSAR_2 nd _DIFF_3 rd _LAG MSAR_2 nd _DIFF_4 th _LAG MSAR_2 nd _DIFF_5 th _LAG;
RUN; QUIT;
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Table 2 SAS Output –An Example of an MSA HPI adjusted returns ADF Test at LEVEL WITH 5 LAGS (Refers to Equation 2)
NULL HYPOTHESIS: ALBANY_R10X (SEASONALLY ADJUSTED RETURNS) HAS A UNIT ROOT 08:37 Monday, June 27, 2005 LAG LENGTH: 5 (FIXED) AUGMENTED DICKEY-FULLER TEST STATISTICS, TEST CRITICAL VALUES: 1% LEVEL T-STATISTICS = -3.54 5% LEVEL T-STATISTICS = -2.91 10% LEVEL T-STATISTICS = -2.59 FIRST DIFFERENCE WITH 5 LAGS
Model: MODEL1 Dependent Variable: MSAR_1ST_DIFF ADJ RETURNS 1ST DIF
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 6 0.04227 0.00705 24.91 <.0001 Error 55 0.01556 0.00028284 Corrected Total 61 0.05783
Root MSE 0.01682 R-Square 0.7310 Dependent Mean 0.00123 Adj R-Sq 0.7016 Coeff Var 1368.64759
Parameter Estimates
Parameter Standard Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 -0.00485 0.00418 -1.16 0.2513 MSAR_LAG1 ALBANY_R10X(-1) 1 -0.62676 0.37449 -1.67 0.0999 MSAR_DIFF_LAG1 D(ALBANY_R10X(-1)) 1 -0.73491 0.35708 -2.06 0.0443 MSAR_DIFF_LAG2 D(ALBANY_R10X(-2)) 1 -0.64765 0.31582 -2.05 0.0451 MSAR_DIFF_LAG3 D(ALBANY_R10X(-3)) 1 -0.50328 0.25901 -1.94 0.0571 MSAR_DIFF_LAG4 D(ALBANY_R10X(-4)) 1 -0.40500 0.19318 -2.10 0.0407 MSAR_DIFF_LAG5 D(ALBANY_R10X(-5)) 1 -0.09738 0.11294 -0.86 0.3923
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Table 3 SAS Output –An Example of an MSA HPI adjusted returns ADF Test First Difference WITH 5 LAGS (Refers to Equation 2)
NUL L HYPOTHESIS: ALBANY_R10X (SEASONALLY ADJUSTED RETURNS) HAS A UNIT ROOT 08:37 Monday, June 27, 2005 LAG LENGTH: 5 (FIXED) AUGMENTED DICKEY-FULLER TEST STATISTICS, TEST CRITICAL VALUES: 1% LEVEL T-STATISTICS = -3.54 5% LEVEL T-STATISTICS = -2.91 10% LEVEL T-STATISTICS = -2.59 FIRST DIFFERENCE WITH 5 LAGS
Model: MODEL1 Dependent Variable: MSAR_2ND_DIFF D(ALBANY_R10X,2)
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 6 0.18777 0.03130 133.21 <.0001 Error 54 0.01269 0.00023494 Corrected Total 60 0.20046
Root MSE 0.01533 R-Square 0.9367 Dependent Mean 0.00039805 Adj R-Sq 0.9297 Coeff Var 3850.70134
Parameter Estimates
Parameter Standard Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 0.00222 0.00198 1.12 0.2662 MSAR_LAG1_DIFF D(ALBANY_R10X(-1)) 1 -6.20111 0.85614 -7.24 <.0001 MSAR_2DIFF_LAG1 D(ALBANY_R10X(-1),2) 1 3.83355 0.77752 4.93 <.0001 MSAR_2DIFF_LAG2 D(ALBANY_R10X(-2),2) 1 2.68385 0.61666 4.35 <.0001 MSAR_2DIFF_LAG3 D(ALBANY_R10X(-3),2) 1 1.71697 0.43108 3.98 0.0002 MSAR_2DIFF_LAG4 D(ALBANY_R10X(-4),2) 1 0.81340 0.25054 3.25 0.0020 MSAR_2DIFF_LAG5 D(ALBANY_R10X(-5),2) 1 0.26645 0.09848 2.71 0.0091
If the null hypothesis of a unit root is accepted, the series is a random walk. The confidence interval is [-2.91, 2.91] at a p-value of 5 percent. At the series’ level (integrated of order 0, I(0)), the null hypothesis of a unit root is accepted at the 5 percent confidence level for 223 MSAs meaning the ADF test statistic falls within the confidence interval [-2.91, 2.91] indicating that the series’ levels are nonstationary. Similarly, the first difference of these 223 nonstationary MSAs house price return series were then tested and 215 were found stationary in the first difference, or integrated of order 1, denoted I(1). The US house price returns were also found to be I(1) such that the ADF test statistic was -0.87 indicating the series is nonstationary in the level and -4.14 indicating the series is stationary in the first-difference (for an MSA example refer to Tables 3 and 4)
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II.C Testing for Cointegration
Modeling the cointegration between nonstationary variables provides one approach for obtaining useful regression results. Despite being individually nonstationary, a linear combination of two or more time series can be stationary. In this case a conintegrating link is said to exist and suggests there is a long run, or equilibrium, relationship between the two variables.
Here the theoretical long-run relationship between the MSA house price returns and US house price returns is the CAPM. The CAPM and Cointegration approaches are combined to address the nonstationarity of the house price return series where the cointegration equation is the CAPM. For MSAs that have stationary cointegration vectors, the MSA and US house price returns share a common random walk component and the regression coefficients are consistent.
The Engle and Granger (1987) two-step procedure is used to test for cointegration between the MSA and US house price returns. In the first step, the CAPM is the cointegration equation such that
MSA(R j) - r 10y = αj + βj(US(R) - r 10y ) + εj (3)
where MSA(R j)-r10y is the excess house price returns for the jth MSA and US(R)-r10y is the excess 6,7 US house price returns. The residual, εj, is the cointegration vector.
In the second step, the ADF is used to test for a unit root in the cointegration vector, or residual (for an MSA example refer to Table 4). For MSAs where εj is stationary, the MSA and US house price returns are cointegrated. The economic significance is that, if the US and MSA house price returns are linked by a long-run relationship, the coefficient of the CAPM regression is valid though slightly biased.
To summarize the Engle and Granger two-step procedure: Step 1 Estimate a “long-run relationship” Yt=aaa + bbbXt and extract the residuals of this regression
PROC REG DATA= REG_SERIES; MODEL MSAHPI_R10X_D11=USHPI_R10X_D11; OUTPUT OUT=RESIDS R=MSA_RESID; RUN ; QUIT ;
Step 2 Apply ADF on the residuals ( eeet) and test for a unit-root in this series: if ( eeet) has a unit root fi reject cointegration
DATA RESID_SERIES; SET RESIDS; e_1 st _LAG = LAG1 (MSA_RESID); e_ 1 st _DIFF = DIFF1 (MSA_RESID);
e_1 st _DIFF_1 st _LAG = DIF1 (LAG1 (MSA_RESID)); e_1 st _DIFF_2 nd _LAG = DIF1 (LAG2 (MSA_RESID)); e_1 st _DIFF_3 rd _LAG = DIF1 (LAG3 (MSA_RESID)); e_1 st _DIFF_4 th _LAG = DIF1 (LAG4 (MSA_RESID)); e_1 st _DIFF_5 th _LAG = DIF1 (LAG5 (MSA_RESID)); RUN ;
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PROC REG DATA = RESID_SERIES; MODEL e_ 1 st _DIFF = e_1 st _LAG e_ 1 st _DIFF e_1 st _LAG e_1st _DIFF_2 nd _LAG e_1st _DIFF_3 rd _LAG e_1st _DIFF_4 th _LAG e_1st _DIFF_5 th _LAG; RUN; QUIT;
Note that according to the way this test is specified, this is a test of no-cointegration, acceptance of a unit root in the residuals suggest that the residual term is non-stationary, which implies rejection of cointegration. This may cause some confusion since acceptance of the Null of a unit root in the residuals suggest rejection of cointegration.
Table 4. SAS Output MSA_HPI = US_HPI Long-Term Relationship’s residuals: Unit Root Test – Level With 5 Lags
NULL HYPOTHESIS: ALBANY_RESID HAS A UNIT ROOT 08:37 Monday, June 27, 2005 LAG LENGTH: 5 (FIXED) AUGMENTED DICKEY-FULLER TEST STATISTICS, TEST CRITICAL VALUES: 1% LEVEL T-STATISTICS = -3.54 5% LEVEL T-STATISTICS = -2.91 10% LEVEL T-STATISTICS = -2.59 FIRST DIFFERENCE WITH 5 LAGS
Model: MODEL1 Dependent Variable: RESID_1ST_DIFF 1ST DIF
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 6 0.04445 0.00741 28.06 <.0001 Error 55 0.01452 0.00026406 Corrected Total 61 0.05898
Root MSE 0.01625 R-Square 0.7537 Dependent Mean 0.00102 Adj R-Sq 0.7269 Coeff Var 1595.06534
Parameter Estimates
Parameter Standard Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 0.00063301 0.00207 0.31 0.7611 RESID_LAG1 ALBANY_RESID (-1) 1 -1.17357 0.43433 -2.70 0.0091 RESID_DIFF_LAG1 D(ALBANY_RESID (-1)) 1 -0.24529 0.40169 -0.61 0.5439 RESID_DIFF_LAG2 D(ALBANY_RESID (-2)) 1 -0.25782 0.33927 -0.76 0.4505 RESID_DIFF_LAG3 D(ALBANY_RESID (-3)) 1 -0.21881 0.26759 -0.82 0.4171 RESID_DIFF_LAG4 D(ALBANY_RESID (-4)) 1 -0.23572 0.19377 -1.22 0.2290 RESID_DIFF_LAG5 D(ALBANY_RESID (-5)) 1 -0.03257 0.11007 -0.30 0.7684
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II.D Error Correction Mechanism
The Error Correction Model is used to estimate the acceleration speed of the short-run deviation to the long-run equilibrium. The error correction model is
(4) ∆MSA _ R ,ti = γ 1∆USA _ Rt + γ 2 [MSA _ R ,ti −1 − α i + β iUSA _ Rt −1 ]+ ε ,ti
The residuals of the long-run model and the first differences of the house price return series are used to estimate the error correction model to determine the short-run deviations from equilibrium.
Of particular interest is the coefficient of the error correction term, γ2, which indicates the speed at which the series returns to equilibrium. For values of γ2 that are negative (positive) and less than (equal to) zero, the series converge to (diverge from) the long-run equilibrium (Refer to Table 5). Residuals ( eeet) estimated from the “long-run relationship” Yt=aaa + bbbXt used in the new regression such as:
PROC REG DATA = RESID_SERIES; *MSA, US RETURNS AND RESIDUALS 1 ST DIFFERENCE, NO INTERCEPT; MODEL MSAR_1 st _DIFF = USR_1 St _DIFF e_1st _LAG /NOINT ; RUN ; QUIT ;
Table 5 – Residuals ( eeet) estimated from the “long-run relationship” Yt=aaa + bbbXt used in the regression (ECM) – Notice that there is no intercept in this model
Model: MODEL1 Dependent Variable: MSAR_1ST_DIFF ADJ RETURNS 1ST DIF
NOTE: No intercept in model. R-Square is redefined.
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 2 0.06102 0.03051 80.47 <.0001 Error 65 0.02464 0.00037913 Uncorrected Total 67 0.08566
Root MSE 0.01947 R-Square 0.7123 Dependent Mean 0.00039054 Adj R-Sq 0.7035 Coeff Var 4985.65898
Parameter Estimates
Parameter Standard Variable Label DF Estimate Error t Value Pr > |t|
USAR_ 1St _DIFF ADJ RETURN 1ST DIF 1 0.65127 0.35254 1.85 0.0692 e__LAG1 RESID(-1) 1 -1.44298 0.11375 -12.69 <.0001
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III EMPIRICAL FINDINGS
This paper provides a methodology for estimating consistent MSA-Betas that measure systemic risk of house price returns by geographic location. It uses the financial framework of the CAPM model as the theoretical bases of a Cointegration Equation. The analysis is extended to include the ECM to explore whether short-run deviation in long-run equilibrium may play a role in the systemic risk for the house asset.
We use regression analysis to estimate MSA-Betas based on time series data. To avoid spurious coefficients, the CAPM is coupled with cointegration to provide consistent MSA-Betas for those MSAs where the cointegration vectors (i.e., those MSA returns that are cointegrated with the US returns) are stationary. The CAPM theory implies that the MSA risk premium on house price returns is a product of the US returns and the MSA-Beta (also known as the MSAs’ systemic or market risk). The MSA-Beta is related to the covariance between the MSA’s house price returns and the US returns. When the MSA-Beta is near one, the expected return is the same as the market. For MSA-Betas less than (greater than) one, expected returns are lower (higher) than house price returns for the US.
Consistent MSA-Betas are found for 73 MSAs (refer to Table 6 for results on selected MSAs) . The largest MSA-Beta is 2.29 for New Bedford, MA. This means that for a one percent increase in expected returns for the US, the expected returns for New Bedford increase 2.29 percent. In contrast, Indianapolis, IN, has an MSA-Beta of 0.38. This indicates that a one percent increase in US house price returns results in only a 0.38 percent increase in Indianapolis house price returns.
The analysis is extended to include the ECM. As expected all coefficients on the error term are negative. The coefficient is referred to as the speed-of-adjustment factor. It measures the short-run deviation from the long run equilibrium. As coefficient values near zero, the paths are slow to adjust back to the long-run equilibrium. As they near or exceed one, short-run deviation follow rapid paths to the long-run equilibrium. In general, we find MSAs with rapid acceleration paths have relatively low MSA-Betas, or systemic risk. Conversely, MSAs with slow paths tend to have high systemic risk.
11 NESUG 18 Analysis
Table 6. Model: MSA(R) - rf =a + B*USA(R) – rf Parameters and Performance statistics for selected MSA Series Error Speed Term Beta of SOA Adjustment MSA Name ADF I(0) ADF I(1) t-stat Beta t-stat R2 (SOA) t-stat Philadelphia, PA-NJ -1.14 -3.38 -3.06 1.38 11.00 0.63 -0.13 -2.36 Seattle-Bellevue-Everett, WA -2.68 -3.60 -2.90 0.29 1.39 0.03 -0.18 -2.65 San Luis Obispo-Atascadero-Paso Robles, CA -1.52 -3.66 -2.98 1.75 6.84 0.40 -0.20 -2.84 Providence-Fall River-Warwick, RI-MA -1.34 -3.73 -4.00 2.16 12.41 0.69 -0.21 -2.89 Monmouth-Ocean, NJ -1.60 -3.22 -4.37 2.15 11.68 0.66 -0.24 -3.17 Allentown -Bethlehem-Easton, PA -2.21 -4.30 -3.26 1.16 6.11 0.35 -0.26 -3.24 Hartford, CT -2.67 -4.28 -4.24 1.83 8.66 0.52 -0.32 -3.75 Houston, TX -2.85 -4.91 -4.11 0.62 3.84 0.17 -0.36 -3.93 Springfield, MA -2.22 -4.04 -3.87 1.83 10.16 0.60 -0.37 -3.98 New York, NY -1.65 -3.23 -3.78 1.81 11.99 0.67 -0.39 -4.91 Bergen-Passaic, NJ -1.51 -3.47 -4.05 1.84 11.50 0.65 -0.39 -4.40 Nassau-Suffolk, NY -1.11 -3.13 -3.16 1.91 13.57 0.72 -0.41 -4.83 Trenton, NJ -2.38 -3.76 -5.03 1.83 9.45 0.56 -0.43 -4.33 Bridgeport, CT -1.51 -4.24 -4.07 2.10 10.09 0.59 -0.46 -4.64 Danbury, CT -1.43 -3.33 -3.85 1.77 8.82 0.53 -0.47 -4.60 New Haven-Meriden, CT -2.31 -4.44 -4.96 2.13 9.14 0.54 -0.54 -5.07 Melbourne-Titusville-Palm Bay, FL -0.45 -6.21 -3.08 1.24 8.69 0.52 -0.61 -5.46 Newburgh, NY-PA -1.74 -3.75 -3.62 1.93 7.89 0.47 -0.71 -6.30 Worcester, MA-CT -1.33 -4.49 -3.23 1.96 12.58 0.69 -0.76 -7.11 Orlando, FL -0.41 -5.78 -3.78 1.10 12.14 0.68 -0.81 -6.90 Indianapolis, IN -2.31 -6.74 -3.53 0.38 7.89 0.47 -0.81 -7.24 Sarasota-Bradenton, FL 0.53 -5.24 -2.97 1.11 9.27 0.55 -0.83 -6.83 Manchester, NH -1.71 -4.46 -3.21 2.23 11.74 0.66 -0.85 -7.06 New London-Norwich, CT-RI -1.93 -4.98 -3.29 1.88 8.20 0.49 -0.86 -7.43 Portland, ME -1.21 -4.44 -2.92 1.75 11.76 0.66 -0.86 -7.24 Rochester, NY -1.98 -5.14 -3.15 0.80 9.58 0.57 -0.88 -7.41 Bismarck, ND -2.44 -8.20 -3.03 0.43 2.73 0.10 -0.93 -7.51 Pittsburgh, PA -2.37 -5.80 -2.94 0.51 5.03 0.27 -0.95 -7.92 Eau Claire, WI -2.61 -7.52 -3.27 0.48 2.16 0.06 -0.96 -8.18 Grand Junction, CO -2.82 -6.69 -2.94 0.43 2.24 0.07 -1.02 -8.64 Cincinnati, OH-KY-IN -1.49 -5.26 -3.04 0.45 8.93 0.53 -1.02 -8.32 Rochester, MN -1.94 -6.27 -2.92 0.69 4.11 0.19 -1.04 -9.28 Vineland-Millville-Bridgeton, NJ -2.71 -6.32 -2.99 1.23 3.76 0.18 -1.19 -9.26 Fort Myers-Cape Coral, FL -0.95 -7.78 -3.02 1.08 6.07 0.34 -1.21 -10.40 Ocala, FL -1.66 -6.31 -3.31 1.11 4.29 0.22 -1.22 -9.95 Binghamton, NY -2.33 -5.25 -2.96 0.84 3.20 0.13 -1.24 -9.87 Fitchburg-Leominster, MA -1.32 -4.97 -2.99 2.19 9.02 0.54 -1.25 -10.65 Johnstown, PA -2.01 -7.58 -3.13 0.45 0.96 0.02 -1.31 -8.44 New Bedford, MA -1.07 -5.49 -5.40 2.29 8.01 0.49 -1.41 -12.46
12 NESUG 18 Analysis
End Notes
1 Since March 1996, the Office of Federal Housing Enterprise Oversight (OFHEO) has estimated and published house price indices (HPI). HPI are available for 331 metropolitan statistical areas and the entire US. MSA specific house price returns are computed based on changes in quarterly house prices.
2 Engle, R. F. and Granger, C. W. J. (1987) Cointegration and Error Correction: Representation, Estimation and Testing, Econometrica 55, 251-276.
3 The approach for estimating the indices is based on a modified version of the weighted repeat- sales method (RSM) that uses information on the same properties at two points in time. Since repeat transactions on the same property controls for differences in the quality, the RSM produces constant-quality house price indices.
4 Catherine C. Hood, “SAS Programs to Get the Most from X-12-ARIMA's Modeling and Seasonal Adjustment Diagnostics” SUGI 25
5 Lucas Critique (Lucas 1978)
6 The CAPM provides the theoretical framework for the cointegration equation and estimating the cointegration vector. The capital asset pricing model applied to house price returns is
E(rj ) = rf + [E(rm ) − rr ]β j where
cov( rj ,rM ) β j = 2 σ (rM )
rj= the return on house price for the jth MSA rM=return on house price for the national (market) portfolio rf = (1+r fr ) where r fr is the risk less rate of return
The risk premium on the jth asset depends on the risk premium on the market portfolio, M, and the beta coefficient, βj. The beta, βj, in this research is defined as the covariance between house price return in MSA j and US divided by the variance of returns for the US.
7 Sharpe (1964) discusses the linear relationship between the excess expected return of a security and the market. This relationship can be empirically tested using regression analysis such that
rj = α j + β j (RM − rf ) + e j
where R j is the return on MSA j (i.e., R j=r j ), R M is the excess return on market portfolio (i.e., R M =r M -rf), and e j is the residual for the jth MSA. The coefficients αj is analogous to the risk-free rate and βj represents the beta factor.
13 NESUG 18 Analysis
References Bails, Dale G. and Larry C. Peppers (1982) Business Fluctuations: Forecasting Techniques and Applications , Englewood Cliffs NJ: Prentice-Hall Inc.
Box, George E.P. and Gwilym M Jenkins (1970) Time Series Analysis: Forecasting and Control . San Francisco: Holden-Day.
Dickey, David A. and Wayne A. Fuller (1981), “Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root”, Econometrica 49, 1057-1072.
Engle, Robert F. and Clive W. J. Granger (1987) “Cointegration and Error Correction: Representation, Estimation and Testing,” Econometrica 55, 251-276.
Hoff, David (1983) “The Sharp Form of Oleinik’s Entropy Condition in Several Space Variables,” Transactions of the American Mathematical Society , Vol. 276, No. 2 pp. 707-714
Lintner, John (1965) “The Valuation of Risks Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics (February).
Lucas, Jr., Robert E (1978) “Asset Prices in an Exchange Economy” Econometrica Vol. 46, No. 6 pp. 1429-1445
Mossin, Jan (1966) “Equilbrium in a Capital Asset Market,” Econometrica (October).
Newbold, P. and Clive W. J. Granger (1974) “Experience with Forecasting Univariate Time Series and the Combination of Forecasts, Journal of the Royal Statistical Society. Series A Vol. 137, No. 2 , pp. 131-165
Pankratz, A. (1983). Forecasting with univariate Box-Jenkins models: concepts and cases. New York, J. Wiley and Sons.
Sharpe, William F., (1963) “A Simplified Model of Portfolio Analysis,” Management Science, (January)
Sharpe, William F., (1964) “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk” Journal of Finance 19, 425-442.
Acknowledgements Our sincere thanks to everyone we have had the pleasure of exchanging Time Series analysis related ideas with in recent years. Special thanks to Ian Keith, Staff Analyst, Titan Corp - U.S. Department of Housing and Urban Development, his constructive suggestions added much to this paper. Trademarks SAS ® and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the US and other countries. ® Indicates US registration. Contact Information The authors welcome and encourage any questions, corrections, improvements, feedback, remarks, both on- and off-topic via email at: Ismail Mohamed, Sr. Software Engineer, TITAN Corp - U.S. Department of Housing and Urban Development E-mail: [email protected] Phone: 202-708-1464 x5884; Theresa DiVenti, Sr. Economist, U.S. Department of Housing and Urban Development, 451 7 th Street, SW, Room 8212, Washington, DC 20410; E-mail: [email protected] Phone: 202-708-1464 x5883
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