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Chapter 3 Prediction and model selection

Chapter 3 Prediction and model selection. Chapter 3. Contents. 3.1.  Properties of MMSE of prediction. 3.2.  The computation of ARIMA forecasts. 3.3.  Interpreting the forecasts from ARIMA models. 3.4.  Prediction confidence intervals. 3.5. Forecasting updating.

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Chapter 3 Prediction and model selection

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  1. Chapter 3 Prediction and model selection Introduction to time series (2008)

  2. Chapter 3. Contents. 3.1.  Properties of MMSE of prediction. 3.2.  The computation of ARIMA forecasts. 3.3.  Interpreting the forecasts from ARIMA models. 3.4.  Prediction confidence intervals. 3.5. Forecasting updating. 3.6. Model selection criteria. Introduction to time series (2008)

  3. We assume an observed sample: • and want to generate predictions of future values given the observations, • T is the forecast origin and k the forecast horizon. Introduction to time series (2008)

  4. Three components • 1. Estimation of new values: prediction. • 2. Measure of the uncertainty: prediction intervals. • 3. Arrival of new data: updating. Introduction to time series (2008)

  5. Chapter 3. Prediction and model selection. 3.1. Properties of MMSE of prediction. Introduction to time series (2008)

  6. Properties of MMSE of prediction • Prediction by the conditional expectation • We have T observations of a zero mean stationary time series , and we want to forecast the value of . • In order to compare alternative forecasting procedures, we need a criterion of optima- lity. Introduction to time series (2008)

  7. Properties of MMSE of prediction • Minimum Mean Square Error Forecasts (MMSF). Forecasts that minimize this criterion can be computed as follows. • Let be the forecast we want to generate, this forecast must minimize • Where the expected value is taken over the joint distribution of and Introduction to time series (2008)

  8. Properties of MMSE of prediction • Using the well-known property of • we obtain Introduction to time series (2008)

  9. Properties of MMSE of prediction • and taking the derivative, we obtain • This result indicates that, conditioning to the observed sample, the MMSEF is obtained by computing the conditional expectation of the random variable given the available information. Introduction to time series (2008)

  10. Properties of MMSE of prediction • Linear predictions • Conditional expectations can be, in some cases, difficult to compute. • Restrict our search to forecasting functions that are linear functions of the observations. • General equation for a linear predictor • . Introduction to time series (2008)

  11. Properties of MMSE of prediction • calling MSEL to the mean square error of a linear forecast • minimizing this expression with respect to the parameters, we have Introduction to time series (2008)

  12. Properties of MMSE of prediction • Which implies that the best linear forecast must be such that the forecast error is uncorrelated with the set of observed variables. • This property suggests the interpretations of the linear predictor as projections. Introduction to time series (2008)

  13. Properties of MMSE of prediction • that is, finding the coefficients of the best linear predictor is equivalent to regress, • then, • where is the covariance matrix of and is the covariance vector between and Introduction to time series (2008)

  14. Chapter 3. Prediction and model selection. 3.2. The computation of ARIMA forecasts. Introduction to time series (2008)

  15. The computation of ARIMA forecasts • Suppose we want to forecast a time series that follows an ARIMA(p,d,q) model. First, we will assume that the parameters are known and the prediction horizon is 1 (k=1) • where h=p+d Introduction to time series (2008)

  16. The computation of ARIMA forecasts • The one-step-ahead forecast will be, • and because the expected value for the observed sample data or the errors are themselves, and the only unknown is Introduction to time series (2008)

  17. The computation of ARIMA forecasts • Therefore, the one-step prediction error is, • remember this is considering that the parameters are known, and therefore, the innovations are also known because we can compute them recursively from the observations Introduction to time series (2008)

  18. The computation of ARIMA forecasts • Multiple steps ahead forecast. • where Introduction to time series (2008)

  19. The computation of ARIMA forecasts • This expression has two parts: • The first one, which depends on the AR coefficients, will determine the form of the long run forecast (eventual forecast equation). • The second one, which depends on the moving average coefficients, will dissapear for k>q Introduction to time series (2008)

  20. The computation of ARIMA forecasts • AR(1) model • for large k, the term , and therefore, the long-run forecast (for any ARMA(p,q)) will go to the mean of the process. Introduction to time series (2008)

  21. The computation of ARIMA forecasts • Random walk with constant. • The forecasts follow a straight line with slope c. If c=0, all forecasts are equal to the last observed value. Introduction to time series (2008)

  22. Chapter 3. Prediction and model selection. 3.3. Interpreting the forecasts from ARIMA models. Introduction to time series (2008)

  23. Interpretation of the forecasts Nonseasonal models. • The eventual forecast function of a nonseasonal ARIMA model verifies for k>q • where Introduction to time series (2008)

  24. Interpretation of the forecasts • Espasa and Peña (1995) proved that the general solution for this equation can be written as, • where, the permanent component is, Introduction to time series (2008)

  25. Interpretation of the forecasts • and the transitory component is, • Permanent component will be given by • With determined by the mean of the stationary process Introduction to time series (2008)

  26. Interpretation of the forecasts • whereas the rest of the parameters, depend on the initial values and change with the forecast origin. • Examples: Introduction to time series (2008)

  27. Interpretation of the forecasts • 1. will be constant for all horizons. • 2. deterministic linear trend with slope , if , then the permanent component is just a constant. • 3. the solution is a quadratic trend with the leading term determined by . If the equation reduces to a linear trend, but now the slope depends on the origin of the forecast. Introduction to time series (2008)

  28. Interpretation of the forecasts • In summary, the long-run forecast from an ARIMA model is the mean if the series is stationary and a polynomial for nonstatio-nary models. • In this last case, the leading term of the polynomial is a constant (when the mean is zero), whereas it depends on the forecast origin (adaptative) if the mean is different from zero. Introduction to time series (2008)

  29. Interpretation of the forecasts • Transitory component. Can be given by • where are the roots of the AR polyno-mial and are coefficientes depending on the forecast origin. Introduction to time series (2008)

  30. Interpretation of the forecasts • Example. Consider the model, • then , and the forecasts must have the form, • where ,the constant that appears as the solution of and , the constant in the transitory equation must Introduction to time series (2008)

  31. Interpretation of the forecasts • be determined from the initial conditions and can be obtained by • and the solution of these two equations is Introduction to time series (2008)

  32. Interpretation of the forecasts • and, • these results indicate that the forecasts are slowly approaching the long run forecast • note that as goes to zero, the adjust-ment made by the transitory decreases exponentially. Cases for Introduction to time series (2008)

  33. Interpretation of the forecasts • Seasonal models. For seasonal processes the forecast will satisfy the equation • Let us assume that D=1, then the seasonal difference Introduction to time series (2008)

  34. Interpretation of the forecasts • and therefore, • which has the property that all the operators involved do not share roots in common. The solution is given by Introduction to time series (2008)

  35. Interpretation of the forecasts • Permanent component has been splitted into two terms, trend component • and the seasonal component Introduction to time series (2008)

  36. Interpretation of the forecasts • Finally the transitory component, which will die out for large horizon is, • the trend component has the same form as for nonseasonal data, but the order is d+1 and therefore the last term is Introduction to time series (2008)

  37. Interpretation of the forecasts • the seasonal component will be given by • and the solution of this equation is a function of period s and values summing zero each s lags. The coefficients are called seasonal coefficients and depend on the forecasting origin. Introduction to time series (2008)

  38. Interpretation of the forecasts • Example: the airline model. • The equation of the forecast is Introduction to time series (2008)

  39. Interpretation of the forecasts • This equation can be written, • that is a linear trend plus a seasonal compo-nent with coefficients that are changing over time. In order to determine the parameters, we need 13 initial conditions Introduction to time series (2008)

  40. Interpretation of the forecasts • With , we obtain that the slope is, • and calling Introduction to time series (2008)

  41. Interpretation of the forecasts • we have that • The seasonal ceofficients are • and will be given by the deviations of the forecast from the trend component. Introduction to time series (2008)

  42. Prediction confidence intervals • Known parameter values. Let us write • then, we can write • taking expected values conditional to data Introduction to time series (2008)

  43. Prediction confidence intervals • The forecast error is • with variance • this equation indicates that the uncertainty of the long run forecast is different for stationary and nonstationary models. Introduction to time series (2008)

  44. Prediction confidence intervals • For a stationary model the series converge since • For an AR(1) model, for instance • The long run forecast goes to the mean, and the uncertainty is finite. Introduction to time series (2008)

  45. Prediction confidence intervals • When the model is nonstationary, the variance of the forecast grows without bounds. This means that we cannot make useful long run forecasts. • If the distribution of the forecast error is known, we can compute confidence intervals for the forecast. Introduction to time series (2008)

  46. Prediction confidence intervals • Assuming normality, the 95% confidence interval for the random variable • We may also need the covariances, for h>0 Introduction to time series (2008)

  47. Prediction confidence intervals • Unknown parameter values. It can be shown that the uncertainty introduced in the forecast for this additional source is small for moderate sample size, and can be ignored in practice. • Suppose an AR(1) model, Introduction to time series (2008)

  48. Prediction confidence intervals • the true forecast error , is related to the observed forecast error • assuming that is fixed, and using that Introduction to time series (2008)

  49. Prediction confidence intervals • we have that • This equation indicates that the forecast error has two components: • uncertainty due to the random behavoir of the observation. • The second measures the parameter uncertainty becuse the parameter are estimated from the sample. (order 1/n - can be ignored for large n). Introduction to time series (2008)

  50. Chapter 3. Prediction and model selection. 3.5. Forecasting updating. Introduction to time series (2008)

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