Time Series Analysis

http://www2.chass.ncsu.edu/garson/pa765/time.htm

Identification of ARIMA parameters:

• Autoregressive component (p). Usually 0, 1, or 2, A value of p=0 means the raw data have no autocorrelation, p=1 means current observations of the series are correlated with themselves at lag 1 (the most common situation), p=2 means correlation at lag 2 also, and so on. An autoregressive component of p = 2 thus means that the dependent (the time series value) is affected by the preceding two values, x_t-1 and x_t-2 independently.

• Integrated component (d). Usually 0, 1, or 2. The d (integrated) component is simply 0 if the raw data are stationary to begin with, 1 if there is a linear trend (the usual case), or 2 if there is a quadratic trend. Higher positive values are possible but very rarely useful. An ARIMA (0,1,0) model is a random walk model in which differencing can be used to remove the linear trend but the remaining variation cannot be explained either on an autoregressive nor on a moving average basis.

• Moving average component (q). Usually 0, 1, or 2. A value of q=0 means there are no shocks in the series and the series is purely an autoregressive one. A setting of q=1 means current observations are correlated with shocks at lag 1, q=2 means they are correlated with shocks at lag 2, and so on. Normally the researcher will set either p or q to a positive value but not both as that may cause overfitting the solution to noise in the data.

  The values of the p and q parameters may be inferred by looking at autocorrelation and partial autocorrelation functions as discussed below.

• Constants: When d=0 there is usually a constant term equal to the mean of the time series. When d=1 there is usually a constant term reflecting the non-zero average trend. When d=2 there is normally no constant; if a constant is added, then it reflects the value of the "trend in the trend."

• There can also be components reflecting continuous variables entered as independents in the ARIMA model.

Autocorrelation and partial autocorrelation functions (ACF and PACF) can also be used to estimate p and q. Specifically, ACF and PACF plots plot deviations from zero autocorrelation by time period: the larger the positive or negative autocorrelation for a period, the longer the plot line to the right (positive) or left (negative) of zero. ACF and PACF are obtained in SPSS under Graphs/Time Series/Autocorrelations.

• Autoregressive models. AR models are indicated when PACF cuts off sharply at lag x but ACF declines slowly. To determine tentatively the value of p, look at the PACF plot and determine the highest lag at which the PACF is significant.

• Moving average models. MA models are indicated by a rapidly declining ACF and PACF. If the ACF does not decline slowly but rather cuts off sharply at lag x, this is suggests setting q=x, thereby adding a moving average component. If autocorrelation is negative at lag-1 then this also indicates the need for an MA (q) term higher than 0.

Other rules of thumb:

• ARIMA (p,0,0): ACF is spiked at lag p and declines toward 0. PACF is spiked at lag 1 to lag p.
• ARIMA (0,1,0): Random walk model. The only effect is a non-seasonal differencing to remove a linear trend. ACF is either constant or is balanced between positive and negative. PACF is spiked only at lag 1.
• ARIMA (1,1,0): First-order autoregressive model. There is non-seasonal differencing to remove a linear trend, and lagging the dependent variable by 1.
• ARIMA (0,1,1): Simple exponential smoothing model. There is non-seasonal differencing to remove a linear trend, and lagging shock effects by 1.
• ARIMA (0,0,p): ACF is spiked at lags 1 to p, declining sharply thereafter to 0. PACF is spiked at lags 1 to p, declining more slowly toward 0.
• ARIMA (p,0,q): ACF and PACF both decline slowly toward 0. PACF declines erratically due to shock effects.
• ARIMA (1,1,1): A mixed model. Warning: normally one does not include both autoregressive effects and moving average effects in the same model because this may lead to overfitting the data to noise, and may reduce the reliability of interpretation of the significance of individual components in the model.

As a rule of thumb, one may wish to start with p=1 and/or q=1 and then increase the p and/or q values if the ACF and PACF for the residuals display spiking.

I have learned the method.Thanks