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Using SAS® for Cointegration and Error Correction Mechanism Approaches: Estimating a Capitla Asset Pricing Model (CAPM) For

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Using SAS® for Cointegration and Error Correction Mechanism Approaches: Estimating a Capitla Asset Pricing Model (CAPM) For 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. 2 NESUG 18 Analysis 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.
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