Ch8 Time Series Modeling 1. Autoregressive Integrated Moving Average Model (ARIMA Model), popularly known as the Box-Jenkins Methodology 2. Vector Autoregression Model (VAR Model) 3. Autoregressive Conditional Heteroscedasticity (ARCH Model) or Generalized Autoregressive Conditional Heterscedasticity (GARCH Model)

For stationary process: 1. Moving average models 1. ARIMA Model Time series models “explain” the movement of a time series by relating it to its past values and to a weighted sum of current and lagged random disturbances. For stationary process: 1. Moving average models 2. Autoregressive models 3. Autoregressive-moving average models For non-stationary process: 4.Autoregressive integrated moving average models

Moving Average Model: A Special Case of ARIMA Moving average process of order q: MA(q) Sample autocorrelation function can be used to specify the order of moving average process.

Autoregressive Model: A Special Case of ARIMA Autoregressive models: AR(p) Sample partial-autocorrelation function can be used to specify the order of autoregressive process.

Mixed Autoregressive-Moving Average Models (ARMA): A Special Case of ARIMA Mixed autoregressive-moving average process of order (p,q): ARMA(p,q)

Non-stationary processes For non-stationary series, we can adopt ARIMA(p,d,q) models . Firstly, we can use ADF test or directly observe its autocorrelogram to determine the degree of homogeneity d. After d is determined, we can work with the stationary series , and examine both its autocorrelation function and its partial autocorrelation function to determine possible values for p and q.

The Box-Jenkins (BJ) Methodology Step 1. Identification Chosing p,d,q with the help of correlogram and ADF test Step 2. Estimation Parameter estimation of the chosen model Step 3. Diagnostic checking Are the estimated residuals white noise? Step 4. Forecasting.

2. Vector Autoregression Model (VAR): The Sims Methodology where the u’s are the stochastic error terms, called impulses or innovations or shocks in the language of VAR.

2. VAR vs. Structural Model A structural model means that the specific relationships between variables are based ( either formally or informally) on economic and finance theories A VAR makes minimal theoretical demands on the structure of a model. With a VAR, one needs to specify only two things: (1) the variables (endogenous and exogenous) and (2) the largest number of lags.

2. Technical Problems of VAR A purist may argue all variables are assumed to be endogenous. Both endogenous and exogenous variables assume have the same lag length. One way of deciding the lag length is to use a criterion like the Akaike or Schwaz and choose that model gives the lowest values of these criteria. There is no question that “trial and error” is inevitable.

2. Advantages of VAR VAR is introduced as an alternative approach to multi-equation modeling through the work of C. A. Sims (1980) VAR provides a framework for testing Granger causality between each set of variables VAR is useful for understanding the relationships between several series.In a VAR model, the explanatory variables might influence the dependent variable, but there is no possibility that the dependent variable influences the explanatory variable. VARs usually have better forecasting ability than sophisticated economic/financial models.

2. Drawbacks of VAR VAR models are atheoretical; that is, do not draw heavily on economic and finance theories VAR has a lag length selection problem.One must trade off having a sufficient number of lags and having a sufficient number of free parameters. In practice, one often finds it necessary to constrain the number of lags to be less than what is ideal given the nature of the dynamics. Because of its emphasis on forecasting, VAR models are less suited for policy analysis.

2. VAR and VECM If all the variables in the VAR are stationary, OLS can be used to estimate each equation and standard statistical methods can be employed. If the variables under study have unit roots and are cointegrated, a variant on the VAR called the vector error correction model, or VECM, should be used.

3. ARCH and GARCH Models ARCH: autoregressive conditional heterscedasticity ARCH models were introduced by Engle (1982) and generalized as GARCH (Generalized ARCH) by Bollerslev (1986). ARCH and GARCH are specially designed to model and forecast conditional variances. The variance of the dependent variable is modeled as a function of past values of the dependent variable and independent, or exogenous variables. These models are widely used in financial time series analysis.

3. Significance of ARCH and GARCH Models There are several reasons that you may want to model and forecast volatility. You may want to analyze the risk of holding an asset or the value of an option; Forecast confidence intervals may be time-varying, so that more accurate intervals can be obtained by modeling the variance of the errors; More efficient estimators can be obtained if heteroscedasticity in the errors is handled properly; These models are also consistent with the volatility clustering often seen in financial time series such as returns data where large changes in returns are likely to be followed by further large changes.

3. The ARCH Specification In developing an ARCH model, you will have to consider two distinct specifications:one for the conditional mean and one for the conditional variance. The conditional mean is the expected value of a random variable when the expectation is influenced (conditioned) by knowledge of other random variables. The mean is usually a function of these other variables. Similarly, the conditional variance is the variance of a random variable conditioned by knowledge of other random variable.

3. An Example: The GARCH (1,1) Model In the standard GARCH (1,1) specification: In fact, this is by far the most popular specification.

3. An Example: The GARCH (1,1) Model, continued The mean equation given in (1) is written as a function of exogenous variables with an error term. The objective of modeling the conditional mean is to construct a series of squared residuals from which to derive the conditional variance. From a time series of the squared residuals of the conditional mean equation we develop the equation for the conditional variance. Since is the one-period ahead forecast variance based on past information, it is called the conditional variance. The conditional variance equation specified in (2) is a function of three terms:

3. An Example: The GARCH (1,1) Model, continued The (1,1) in GARCH (1,1) refers to the presence of a first-order GARCH term ( the first term in parentheses) and a first-order ARCH term ( the second term in parentheses). An ordinary ARCH model is a special case of a GARCH specification in which there are no lagged forecast variances in the conditional variance equation.

3. GARCH Specification In general, we could have any number of ARCH terms and any number of GARCH terms. The GARCH (p,q) model refers to the following equation for The above model can be generalized even further by including one or more exogenous or predetermined variables as additional determinants of the error variance

3.The ARCH-M Model The X’s in equation (1) represents exogenous or predetermined variables that are included in the mean equation. If we introduce the conditional variance into the mean equation, we get the ARCH-in-Mean (ARCH-M). A variant of the ARCH-M specification uses the conditional standard deviation in place of the conditional variance.

3. The ARCH-M Model, continued The ARCH-M model is often used in financial applications where the expected return on an asset is related to the expected asset risk. The estimated coefficient on the expected risk is a measure of the risk-return tradeoff. Note in the conditional mean equation the variance is transformed into a conditional standard deviation so that it is in the same unit of measurement as the risk premium being modeled.