Lecture 7: Forecasting: Putting it ALL together

The full model The model with seasonality, quadratic trend, and ARMA components can be written: Ummmm, say what???? The autoregressive components allow us to control for the fact that data is directly related to itself over time. The moving average components, which are often less important, can be used in instances where past errors are expected to be useful in forecasting. 2

Model selection Autocorrelation (AC) can be used to choose a model. The autocorrelations measure any correlation or persistence. For ARMA(p,q) models, autocorrelations begin behaving like an AR(p) process after lag q. Partial autocorrelations (PAC) only analyze direct correlations. For ARMA(p,q) processes, PACs begin behaving like an MA(q) process after lag p. For AR(p) process, the autocorrelation is never theoretically zero, but PAC cuts off after lag p. For MA(q) process, the PAC is never theoretically zero, but AC cuts off after lag q. 3

Model selection An important statistic that can used in choosing a model is the Schwarz Bayesian Information Criteria. It rewards models that reduce the sum of squared errors, while penalizing models with too many regressors. SIC=log(SSE/T)+(k/T)log(T), where k is the number of regressors. The first part is our reward for reducing the sum of squared errors. The second part is our penalty for adding regressors. We prefer smaller numbers to larger number (-17 is smaller than -10). 4

Important commands in EViews ar(1): Includes a single autoregressive lag ar(2): Includes a second autoregressive lag Note, if you include only ar(2), EViews will not include a first order autoregressive lag ma(1): Includes a first order moving average term. This is not the same as forecasting using an average of recent values 5

Selecting an appropriate time series model, concluded Determine if trend/seasonality is important If it is, include it in your model Estimate the model with necessary trend/seasonal components. Look at the correlogram of the residuals: From the equation dialogue box: View => Residual Tests => Correlogram Q-statistics If ACs decay slowly with abrupt cutoff in PAC, this is indicative of AR components. If the PAC doesn’t cutoff, you may need to include MA components as well. Re-estimate the full model with trend/seasonality included with necessary ARMA components. You will likely have several models to choose from. 6

Selecting an appropriate model, cont. After you estimate each model, record SIC/AIC values Use the SIC/AIC values to select an appropriate model. Finally, investigate the final set of residuals. There should be no correlation in your residuals. Evidenced by individual correlation coefficients within 95% confidence intervals about zero. Ljung-Box Q-statistics should be small with probability values typically in excess of

Exponential smoothing Very useful when we have only a handful of observations Exponential smoothing can be modified to account for trend and seasonality. If you suspect your data does not contain trend/seasonality, simply use single exponential smoothing: The forecast h periods into the future is constant and given by: 8

Exponential smoothing with trend Obtain the smoothed level series, L t : L t =  y t +(1-  )(L t-1 +T t-1 ) The trend series, T t is formed as: T t =  L t -L t-1 )+(1-  )(T t-1 ) The forecasted series: 9

Example: Suppose the first three observations of a series are 500, 350, and 250. Suppose  =0.30,  =0.10. Suppose the trend and level are initialized such that T 1 =0, L 1 =500. L 2 =0.30 (350) +(1-0.30)(500-0) = 455 T 2 =0.10( ) (0) = The forecasted value for the third period then becomes: 10

Exponential smoothing: Trend and seasonality Eviews also allows for exponential smoothing with trend and seasonality. With seasonality we use a smoothed series along with estimates based on trends and seasonality. Which option should I select? If you believe your series lacks either seasonality or trend, single smoothing works perfectly. From visual inspection of your series, if only trend appears to be present, you will need either “double smoothing” or “Holt- Winters” with “no seasonal.” If seasonality and trend are expected, you will need to use “Holt-Winters” with the allowance of multiplicative or additive seasonality. 11

HUH? Eviews provides five options when you ask it, no tell it, to provide exponential smoothing: Single: (no seasonality/no trend) Double: (trend – value of  =  ). Holt-Winters – No seasonal (Trend,  and  are not equal, but are estimated in the data). Holt-Winters – Additive (Trend and Seasonality. The seasonal component is estimated with an additive filter). Holt-Winters - Multiplicative 12

Breaks? Uh oh? My data appears to have a break. The developed time series methods assume the black box generating the data is constant. Not necessarily true: Learning curves may cause cost curves to decrease Acquisition of companies or new technologies may alter sales/costs 13

Dealing with breaks? Solutions: Limit the sample to the post break period Sometimes taking logs and/or differencing can help mitigate the effects of breaks/outliers. Include variables that help identify the breaks Model the breaks directly: The most obvious way is to include a break in mean and/or a break in trend. We should make sure that the included break is modeled in a sensible way A negative linear trend, for example, will imply the data may eventually turn negative. 14

Break in mean 15

Break in trend 16

Statistics useful in comparing the out of sample forecasting accuracy Mean squared error: For an h-step extrapolation forecast: Root mean squared error is the square root of this number. Mean absolute error 17

In Eviews: If you have a forecasted series, say xf, and an original series x, you can calculate the mean squared error as: genr To calculate the moving average forecasts: Suppose you use the most recent four periods Limit your data set to include only the last four observations A variable called “maf_4” is calculated by: genr 18