Empirical Techniques in Econometrics - Essay Example

Empirical techniques in Econometrics Introduction: Econometrics is the application of statistical methods for solving the financial issues. It hasmany applications like - the effect of the economic conditions on the financial markets, the asset price derivations, predicting the future financial variables and other financial decision-makings. In econometrics there is a lack of adequate test data for applying the particular methodology, this is termed as the small samples problem. There are further constraints in Econometrics with respect to data revisions and the measurement error. These problems are generally faced due to the subsequent revisions in the reference data and the incorrect data estimation or incorrect measurement of data. The frequency of observation of the financial data has far-reaching implications. For the sake of understanding, just imagine the example of the prices of stocks in the share market, they are highly volatile and keep on changing every day, hour, minutes and so on. So to have precise knowledge of these prices one needs to have large quantum of data, in tens of thousands or in millions. Financial data are very noisy in the sense that it is highly difficult to draw a certain pattern or trend from the available data. In other sense the data doesn't have a specific distribution. But approximations are applied for modeling of the market and for analyzing the future trends, values of financial variables. Types of data: Time Series data: The data is collected over a period of time. There is a specific frequency of observation for such type of data. It can be the data collected over a defined periodicity or a data sampled at specific regularity. Cross sectional Data: It is the data on multiple variables, observed at a specific single point of time. The examples of cross section data are - The stock returns of a stock exchange at any point of time, A sample taken for the bond credit ratings for any bank. Cross sectional data helps to make analysis of several things like - expected returns after investing in the shares of a particular company and its relation with the size of the company, the status of debts of a country and its relation with the GDP. Panel Data: These are data having both the dimensions of time series as well as cross sections, e.g. the weekly prices of mid cap shares over the period of five years. Cointergration: The macroeconomics and financial economics has empirical research based on time series. The macroeconomic time series has a nonstationarity property, which means that the variable doesn't return to a constant value or a linear trend. The stationary processes has a basic tendency of moving around a linear value i.e. the mean value and its fluctuation from this value is termed as the deviation. The variables such as employment, asset prices, gross domestic product follow a nonstationarity property and possess stochastic trends. Consider the trend in the financial return series like the rate of change of daily exchange rate. The figure shows the volatility of returns. Fig.1 Earlier it was a general practice to estimate nonstationary process equations in macroeconomic models by the simple linear regression. Clive Granger (1981) proposed a solution to the time series by a simple regression equation: (1) where, = dependent variable = single exogenous regressor = white noise To stress the solution, Granger defined the degree of integaration of the variable. Suppose a variable can be made nearly stationary by differencing it d times, then it can be termed as integrated of order d or I(d). Stationary random variables are I(0). In equation (1), if I(1) and I(1), then I(1). But there exists an important exception, if I(0) then I(0). The linear combination, holds same statistical properties as an I(0) variable. This makes the above combination unique, so the coefficient is also unique. In such case the variable and are called cointegrated. In generalized way, we can say that the variables are cointegrated if a linear combination of a set of I(1) variables is I(0). The Granger representation theorem marks the importance of cointegration in the modeling of nonstationary economic series. For illustration, consider the bivariate autoregressive system of order p: (2) (3) and are cointegrated; and are white noise. Granger representation theorem states that the nonstationary economic system can be represented as: (4) (5) Here either or deviates from zero. Both the above system equations are balanced, i.e. the order of integration on left hand and right hand are same; as If defines a dynamic equilibrium relationship between the two economic variables y and x, then represents the degree of disequilibrium. Applications of cointegration: It has been used as a common econometric tool in several areas, where current long term interests are predicted via expected short term rates, the expected future income restricts current consumption. Study of exchange rates and prices reveals the way how cointegration makes empirical analysis for investigating old problems. (Cointegration, KUNGL) The regression model of Purchasing Power Parity (PPP) is given as: (6) Where, = logarithm of the exchange rate between home and foreign currency. = logarithm of price level of home goods = logarithm of price level of foreign goods The next level of studies made an assumption that the cointegrating relationship between has a cointegrating vector = (1,-1,1). However with realistic trend, real exchange rate, different prices in trading, the cointegrating factor is defined as , where parameters and , if deviate from unity, reflects different trends for untreaded and treaded goods. The regression model now becomes (7) Estimating the parameters of this equation, we get (8) The estimated residuals of equation (8) are represented in Fig. 2 Fig. 2 It is seen that the estimated residual are fluctuating around zero, which imply the stationarity but the pattern suggest that it is nonstationary. (Engle and Granger, 1987) Capital Asset Pricing Model - CAPM : It provides the means to make predictions about risk measurement and its relation with the risk. There are certain empirical problems associated with the theoretical failures of CAPM. These problems are generally arising because of the difficulties in implementing valid test results of the model. CAPM is built on the basic model developed by Harry Markowitz (1959). In Markowitz model, a portfolio of an investor is selected at time t-1, which gives a stochastic return at time t. It is assumed that the investor is disinclined towards the risk and they are affected only by the mean and variance of the periodic investment returns. Sharpe (1964) and Lintner (1965) made two important assumptions in the Markowitz model to make a portfolio, which must be mean variance efficient. First assumption: complete agreement - the asset prices are given at time t-1, the investor accepts the joint distribution of asset returns from t-1 to t. This distribution is a real distribution, which provides the returns to test the model. Second assumption: borrowing and lending at risk free rate - This assumption is valid for all the investors and doesn't depend on the amount borrowed or lent. With the agreement of distribution of returns, there is same opportunity for all the investors and all of them can have risk free lending or borrowing. In other way round we can say that as per CAPM assumptions, the market portfolio M must have the minimum variance frontier so as to clear the asset market. This implies that the algebraic relation valid for minimum variance portfolio must be valid for the market portfolio. (Minimum Variance Condition for M): (9) In the above equation, = Expected return on asset i = the market beta of asset i, which is defined as the covariance of the asset return with the market return divided by the variance of the market return. Therefore, Market beta, = expected return on assets for which market betas equal to zero, i.e. these are the returns of the asset uncorrelated with the market return. = expected market return. = risk premium. The market beta of asset i measures the sensitivity of the asset's return as per the variation in the market returns. In CAPM, the risk of market portfolio is the ratio of weighted average of covariance risks of different assets in M with the variance of the returns of the asset. Therefore is the covariance risk of asset i in market portfolio M with respect to average covariance risk of assets, that comes to be the variance of the market return. The assumption of risk-free borrowing and lending is to reduce, the expected return for the assets having zero beta value. The market return and the risky asset's return are uncorrelated, therefore its beta has a zero value. A risky asset becomes riskless in the market portfolio as it doesn't add anything in the variance of the market return. As per Sharpe-Lintner, in case of the risk-free borrowing and lending, the expected return of the assets which doesn't have any correlation with the market return, i.e. , must have a value equal to risk free rate, . The Sharpe-Lintner CAPM equation becomes, (10) From the above equation it is seen that the expected return on any asset i is the sum of risk free interest rate, and the risk premium. Risk-free borrowing and lending is an unrealistic assumption. Fischer Black(1972) modified the CAPM version. This version avoids the risk-free borrowing and lending. As per this model, the mean-variance efficient market portfolio can be derived by unrestricted short sales of risky assets. The difference between Sharpe-Lintner and Black versions of CAPM is mainly concerned with the expected return on assets uncorrelated with the market, . As per Black, must have value lesser than the expected market return,, which makes the positive premium for beta. 16 14 12 10 8 6 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Beta Fig. 3 The Sharpe-Lintner CAPM shows that the predicted portfolios follow a straight line. The interception of the straight line represents the risk free rate, and the slope of the straight line equals the expected excess return on the market i.e. (CAPM, Fama and French) The basic structure of CAPM is based on assumptions, which may not hold well in the realistic market, it is just an approximation. The CAPM assumptions are unrealistic. As per CAPM, investors care only about variance and mean of a specific period portfolio return, which is not enough. In reality investors also care about the relationship between their portfolio return and its covariance with the future investment options and labor income. If consideration of these parameters is more appropriate then, market beta doesn't provide complete details of an asset's risk. It is quite obvious that differences in expected return are not completely explained by differences in beta. Merton (1973), proposed Intertemporal Capital Asset Pricing Model (ICAPM), which is an extension of CAPM with different assumptions about investor objectives. In the ICAPM, investors are not only concerned with their one period payoff but also with future opportunities. Fama (1996) indicates that ICAPM generalizes the logic of CAPM on an extended basis. Even with the CAPM assumptions, if risk-free borrowing and lending or risky asset's to be short sale are allowed, the clearing prices of the market indicates that the market portfolio is multifactor efficient. Multifactor efficiency states the relation between beta risks and expected return, but it needs additional betas in addition to market beta to explain the expected returns. (CAPM, Fama and French) Three-Factor Model : Fama and French proposed the three factor model as - , (11) where, = small minus big, it is the difference between the returns on diversified portfolios of small and big stocks. = high minus low, it is the difference between the returns on diversified portfolios of high and low B/M (Book value to Market price ratio) stocks. In this case the betas are slopes in the multiple regression of on , and . The expected return equation in the three-factor model in time-series regression is , (12) where, the intercept is zero for all assets i. This model accommodates most of the market variations in average return for the portfolios formed on book-to-market equity, size and price ratios. The three-factor model is popularly used internationally for empirical research for the expected returns. In this model the main theoretical lacunae is its empirical motivation. The high minus low (HML) and small minus big (SML) are not inspired by predictions of state variables. These terms are additionally introduced just to take care of the average variation in stock returns with size and book-to-market equity ratio. The mean excess return as per beta variation is as shown in Fig. 4 Conclusion : It is seen that the cointegration method helps to predict current long term interests via expected short term rates. It is a basic empirical method used to put financial stock returns, interests in a regression equation. It helps to analyze the nonstationary economic time series. CAPM provides good justification about the average returns on stocks, portfolios, funds and their values higher than others. But the average returns of multiple investment Changing Size within book/market category 1.2 0.9 0.6 0.3 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Predicted Beta Fig. 4 models are not expressible by CAPM. Multifactor models are more effective to determine the average returns on multiple investments. CAPM is better in the measurement of risk and justifies that some assets can earn higher returns than others. Such assets must have higher beta. Beta enhances the average returns. There exists several assets, whose average returns cannot be explained by their beta alone. They need multifactor considerations over CAPM. Multi-factor models are concerned with the high average returns in combination with a tendency to move including other risk factors. In CAPM the basic assumption is that the investor cares only about his investment portfolio which is not the case in reality. The assumptions about risk free borrowing and lending as well as the short sale of assets are not always hold good and does not generate true results. Fama and French (1993, 1996) derived a multifactor model having the market return, returns from small minus big stocks (SMB) and the returns from high minus low book/market stocks. As the Three-factor model is able to take care of other factors which contribute for the return of the market portfolio of an investor than an individual asset. Multi-factor models are more suitable and are popularly used to predict the returns. References Engle, R.F. and Granger, C.W.J : 1987, C0-integration and error-correction: Representation, estimation and testing, Econometrica 55, 251-276 Granger, C.W.J. and Weiss A.A. : 1983, Time-series analysis of error correction models, " Studies in Econometrics, Time series and Multivariate Statistics, in Honour of T.W. Anderson", Academic Press, San Diego, pp 255-278 Fama, Eugene F. and Kenneth R. French. 1996. " Multifactor Explanations of Asset Pricing Anomalies. " Journal of Finance, 51:1, pp. 55-84 Markowitz, Harry. 1959. Portfolio Selection: Efficient Diversification of Investments. Cowels Foundation Monograph No. 16. New York: John Wiley & Sons Inc. Sharpe, William F. 1964. "Capital Asset Prices: A theory of Market Equilibrium under Conditions of Risk." Journal of Finance. 19:3, pp. 425-42 Litner, John 1965. "The Valuation of Risk Assets and the selection of Risky Investments in Stock Portfolios and Capital Budgets. " Review of Economics and Statistics. 47:1, pp. 13-37 Introduction To Econometrics, from http://assets.cambridge.org/97805217/90185/sample/9780521790185ws.pdf Empirical Analysis of Estanbul Stock Exchange http://www.bilkent.edu.tr/economics/papers/06-02%20DP_HakanBerument.doc Fractional Cointegration and Term structure of Interest rates http://www.u-cergy.fr/IMG/2002-28Mignon.pdf Forecasting & Empirical Methods in Finance & Microeconomics http://palm.nber.org/reporter/winter00/diebold.html#N_22_ Cointegration & Autoregressive Conditional Heteroskedasticity http://www.kva.se/KVA_Root/files/newspics/DOC_2003108143127_50163615451_ecoadv03.pdf Capital asset Pricing & Fama-french Three Factor Model http://www.stern.nyu.edu/fin/pdfs/seminars/013w-brennan.pdf Capital asset Pricing Model http://www.investopedia.com/terms/c/capm.asp Fama-French 3 factor Model http://www.investopedia.com/terms/f/famaandfrenchthreefactormodel.asp Capital asset Pricing Model, Famma and French, from http://www-personal.umich.edu/kathrynd/JEP.FamaandFrench.pdf Fama&French3FactorNModel:New Evidence from Emerging Market http://www.csae.ox.ac.uk/conferences/2006-EOI-RPI/papers/csae/Bundoo.pdf Understanding Risk & Returns: CAPM & Fama-French 3 Factor Model http://www.ifa.com/pdf/FamaFrenchThreeFactor_Tuck2003.pdf Statistical Regressional analysis http://www.camo.com/rt/Resources/statistical-regression-analysis.html Multivariate Linear Regression http://www.cab.latech.edu/jcochran/QA610/multivariate%20linear%20regression.ppt Read More