1. 1 Autocorrelation in Time Series data KNN Ch. 12 (pp. 481-501)

2. 2 Causal Models Quantitative Forecasting Time Series Models Regression Exponential Smoothing Trend Models Moving Average Quantitative Forecasting

3. 3 Time Series vs. Cross Sectional Data Time series data is a sequence of observations –collected from a process –with equally spaced periods of time –with equally spaced periods of time. Contrary to restrictions placed on cross- sectional data, the major purpose of forecasting with time series is to extrapolate beyond the range of the explanatory variables.

4. 4 Autoregressive Forecasting

5. 5  The errors u t are independent and normally distributed N(0,  2 )  The autoregressive parameter  has |  | < 1 Regression Model with AR(1) error

6. 6  The previous simple regression model can be expanded to accommodate multiple predictors Multiple Regression Model with AR(1) error

7. 7 Autoregressive expansion  The autocorrelation parameter  is the correlation coefficient between adjacent error terms  Expanding the definition of  t, Autoregressive component Random error component

8. 8 Autoregressive expansion  The correlation coefficient diminishes over time, since |  | < 1  This is why an ACF plot exhibits a diminishing correlation pattern for AR(1) models: ACF PACF

9. 9 Remedial measures for AR errors in regression models  Cochrane – Orcutt procedure  Hildreth – Lu procedure  First differences procedure All estimates are close to each other, the last procedure is the simplest

10. 10 First Differences procedure (regression through the origin) Back transformations:

11. 11 The Blaisdell Company Example (Blaisdell.xls)

12. 12 The Blaisdell Company Example (regression through the origin) Back transformations:

13. 13 Forecasting  Forecasts obtained with autoregressive error regression models are conditional on the past observations  Using recursive relations, two or three-step ahead forecasts can be obtained, but prediction intervals will expand very fast