Terms: Forecast vs Forecast Error We clarify the terms used in the practice problems and the final exam problems. Some statisticians speak of the standard deviation or variance of the forecast. The forecast here is the distribution of future values. It is a random variable, which has a standard error (standard deviation and variance). Other statisticians use the term forecast for the mean of the distribution of future values. The forecast error (the error term in the forecast) is the distribution of future values minus its mean. Using these terms: ! The forecast is a scalar with a non-zero value (unless the forecasted value is zero). ! The forecast error is a random variable with a mean of zero and a non-zero standard error. The final exam problems may use either view. The meaning is clear from the context.
Objectives Actuaries forecast loss costs, mortality rates, reserves, and other values. We want a measure of the quality of the forecast. Illustration: An actuary uses a time series to estimate the average claim severity next year as $10,000. We use this forecast to set rates for auto insurance policies. The procedure used to estimate the future average claim severity may be unbiased, bu the actual claim severity next year will not be exactly $10,000. If the actuary’s estimate is a normal distribution with a mean of $10,000 and a standard deviation of $500, we are 95% confident that the true average claim severity will lie between $9,000 and $11,000. We rely on the actuary’s estimate to set rates for next year. If the actuary’s estimate is a normal distribution with a mean of $10,000 and a standard deviation of $5,000, we are 95% confident that the true average claim severity will lie between zero and $20,000. The actuary’s estimate is less persuasive. To set rates for next year, we may rely on industry averages or the rates of a major competitor. The standard error of the forecast also indicates whether a high or low future value reflects random fluctuation or mistake in the forecast. Illustration: Suppose next year’s average claim severity is $12,000, or $2,000 above the estimate. If the estimated standard deviation is $500, we assume either ! The modeling procedure is not valid. Perhaps average claim severity does not follow this ARIMA process. ! The external environment has changed. Perhaps tort law has been liberalized in the state, and plaintiffs need no longer demonstrate proximate cause. If the estimated standard deviation is $5,000, we assume random fluctuations cause the discrepancy.
Random Variables and Scalars Errors are of three types: model specification error, parameter error, and process error. ! We minimize model specification error and parameter error. ! We measure process error. Model specification error: We may be using the wrong model, such as AR(1) instead of AR(2). The reason for the wrong model might be ! The model was AR(1) in the past, but it changes to AR(2) in the future. ! The model is AR(2), but random fluctuations lead us to believe it is AR(1). This course shows how we choose among models. ! In practice, no model may be correct. We do not presume that interest rates follow a specific ARIMA process. ! The variances of the forecast errors are conditional variances. If the time series is an ARIMA process of the type specified, the variance is … The regression analysis courses discusses ordinary least squares estimators and the time series course discusses additional methods of choosing parameters. More observations in the time series reduce the standard error of the parameters and the forecasts. ! The regression analysis course relates the standard error of the ordinary least squares estimators to the standard error of the forecast. ! The time series course asks: “If the model is correctly specified and the parameters are correct, what is the variance of the forecast?” We can answer this final question most easily by converting the ARMA process to a filter representation, with values...
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