Chapter 1 : Introduction and Review 1.1 What is econometrics? 1.2 Review of linear regression 1.3 Review of univariate non-seasonal ARIMA models

1.1 What is econometrics? Econometrics is the main technique used in this course It is a branch in economics that brings together economic theory and statistics to study economic phenomena. Econometric theory is concerned with the development and extension of statistical techniques appropriate for economic data (e.g. VAR models, Cointegration analysis, etc.)

In this course, we are primarily concerned with applied econometrics. In recent years, a branch in econometrics, has emerged with emphasis on financial applications. Examples of econometric tools in terms of financial applicability : Test whether financial markets are weak-form efficient. Test whether the Capital Asset Pricing Model (CAPM) and the Abitrage Pricing Theory (APT) are appropriate models for the determination of returns on risky assets. Measuring and forecasting the volatility of bond or stock returns (e.g. GARCH models). Modelling long-term relationships between prices and exchange rates (cointegration analysis). Forecasting the correlation between the stock indices of two counties. and so on.

Economic Data vs. Financial Data Financial data and Economic data often differ in terms of their frequency, accuracy, seasonality and other properties. Lack of data is a serious problem for economic data analysis. For example, it might be that the data required on government budget deficits, on population figures, which are measured only on an annual basis. Measurement errors and data revisions are the two other major problems with economic data. Economic data are often estimated based on sample information.

These issues are rarely of concern in finance. Generally the prices and other entities recorded are those at which trades actually took place. Also, financial data are observed at much higher frequencies than economic data. Asset prices are often available at minute-by-minute frequency!!

On the other hand, financial data are often very “noisy” (i. e On the other hand, financial data are often very “noisy” (i.e. it is more difficult to separate the trends and other patterns from the random behaviour). Financial data are almost always not normally distributed, but most techniques in econometrics assume they are. High frequency data often contain additional patterns which are the result of the way the market works (e.g. clustering). These features need to be considered in the model-building process.

1.2 Review of Linear Regression Regression analysis is certainly the single most important technique in econometrics. A statistical technique that attempts to explain movements in one variable (dependent) as a function of movements in a set of other variables (explanatory).

Simple Linear Regression Model Yt = 0 + 1Xt + t ; t ~ N(0, 2) The ’s are the coefficients 0 is the constant (intercept) term in the regression equation. 1 is the slope coefficient. It measures the amount Y will change when X changes by 1 unit. 2 is the variance of t , the random disturbance term.

Examples of regressions used in Financial analysis: How asset returns vary with their level of market risk. Measuring the relationship between spot prices and market risk. Constructing an optimal hedge ratio.

Slope Coefficients Recall that we are trying to explain or predict changes in Y using X. 1 defines the relationship for us. For linear models, the slope is constant.

Linear Regression i.e., the dependent variable has a linear relationship with the explanatory variable cannot use linear regression methods if the equation is not linear in the coefficients

Expected Value of Yt 0 + 1Xt is the deterministic part of the regression equation. i.e. it is the value of Yt determined by a given value of Xt , which we assume to be non-stochastic. can also think of it as the expected value of Yt given Xt E(Yt | Xt) = 0 + 1Xt

Stochastic Error Term (t) Almost always the case that Xt alone cannot explain all the variations in Yt . Can add more variables. But there will still be some variation in Yt that cannot be explained by the model. Sources of error : random shocks, measurement errors etc.

Example : Capital Asset Pricing Model (CAPM) A fundamental idea of modern finance is that an investor needs a financial incentive to take a risk Said differently, the expected return on a risky asset, R, must exceed the return on a safe, or a risk free investment, Rf . Thus the expected excess return, R – Rf , should be positive.

According to the CAPM, the expected excess return on an asset is proportional to the expected excess return on a portfolio of all available assets (the “market portfolio”) i.e. R – Rf = (Rm – Rf ) So, a stock with  < 1 (> )1 has less (more) risk than the market portfolio and therefore has a lower (higher) expected excess return them the market portfolio. The ’s are usually estimated by least squares regression.

The table below gives the estimated ’s for six U.S. stocks Company Estimated  Kellogg (breakfast cercal) 0.24 Waste Management (waste disposal) 0.38 Sprint (long distance telephone) 0.59 Walmat (discount retailer) 0.89 Barnes and Noble (book retailer) 1.03 Best Buy (electronic equipment) 1.80 Microsoft (software) 1.83

Finding the “Best Fit” Recall that we are trying to estimate a linear relationship such that the line passing through the data best represents the underlying true relationship. Difference between the actual values and the estimated values is the residual. Best fit  minimize the residuals.

Ordinary Least Squares A regression estimation techniques that calculates estimates of the slope parameters in our linear regression model so as to minimize the squared residuals :

Goodness of Fit measure : R2 , Adjusted R2 Hypothesis testing : Significance of individual coefficients t test Overall significance of model F test Significance of linear restrictions partial F test Serial correlations Durbin-Watson test (1st order) Lagrange Multipler test (higher order)

1.3 Review of Univariate Non-seasonal ARIMA Models Univariate forecasts based on a statistical analysis of the past data. Differs from conventional regression methods in that the mutual dependence of the observations is of primary interest. Forecasts are linear functions of the sample observations.

Box-Jenkins Methodology Identification of the type of model to be used is a critical step. Differs from other univariate techniques in that a thorough study of the properties of a time series is carried out before applying a forecasting technique.

Principle of Parsimony Find efficiently parameterized models. A model with a large number of parameters will achieve a good historical fit, but post-sample forecasts are likely to be poor. By building a model based on past realizations of a time series we are implicitly assuming that there is some regularity in the process generating the series. One way to view such regularity is through the concept of stationarity. The use of Box-Jenkins modeling techniques requires a stationary process.

A stochastic process is a collection {Xt : t = 1, 2, …, T} of random variables ordered in time. Example : the error term in a linear regression model is assumed to be a stochastic process. A stochastic process is weakly stationary if for all t values E[Xt] =  var(Xt) = 2 cov(XtXt-k) = k  t i.e. its statistical properties do not change over time.

More precisely, a stationary series is one for which the mean and variance are constant across time and the covariance between current and lagged values of the series (autocovariances) depends only on the distance between the time points.

Autocorrelations Covariances are often difficult to interpret because they depend on the units of measurement of the data. Correlations, on the other hand, are scale-free. Thus, we can obtain the same information about the time series by computing the autocorrelations of a time series.

Autocorrelations Coefficient The autocorrelation coefficient between Xt and Xt-k is A graph of the autocorrelations is called a correlogram. Knowledge of the correlogram implies knowledge of the process which generated the series and vice versa.

Partial Autocorrelations Another important function in the Box-Jenkins methodology is the partial autocorrelation function. It measures the strength of the relationship between observations in a series controlling for the effect of the intervening time periods.

Box-Jenkins Methodology Identification. Determine, given a sample of time series observations, what the model of the [stationary] data is. Estimation. Estimate the parameters of the chosen model

Diagnosting checking and Model Selection t-test for the significance of individual coefficients Ljung Box-Pierce (Q) test for the correlations of the residuals Minimize the Akaike Information Criteria and the Schwarz Bayesian Criteria Forecasting with fitted model.

Guidelines for Box-Jenkins Identification Model Correlogram Partial Correlogram AR(p) Dies off Truncates after Lag p MA(q) Truncates after Lag q ARMA(p,q)

Applied Example (Box-Jenkins Modelling of Dow-Jones Industrial Index) DJI index is an index of 30 industrial firms’ stock prices dataset : dji.txt The values in this data set represent monthly averages of the end-of-day values for the index over the period January 1984 to February 1994

SAS Program: Example 1.2 data djm; infile 'd:\teaching\ms4221\dji.txt'; input date:monyy5. djiam @@; format date monyy5.; title 'Dow Jones Index Data'; title2 'Monthly Average'; proc print; run; symbol1 i=join; proc gplot data=djm; format date year4.; plot djiam*date/vminor=1; proc arima data=djm; identify var=djiam; identify var=djiam(1); estimate q=1 method=ml; estimate p=1 method=ml; estimate p=1 q=1 method=ml;