Univariate ARIMA Forecasts (Theory)
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1 Univariate ARIMA Forecasts (Theory) Al Nosedal University of Toronto April 10, 2016 Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
2 All univariate forecasting methods (including ARIMA methods) are based on the same logic. First, the expected value of the time series process is calculated and, second, the expected value is extrapolated into the future. If the current time series observation is Y t, then we are interested in predicting the values of Y t+1, Y t+2,..., Y t+n. We will denote our ARIMA forecast of Y t+n by Y t (n). We call Y t (n) the origin-t forecast of Y with a lead time of n observations. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
3 As a first step in generating an estimate of Y t (n), we calculate the expected value of the Y t process. Our calculations will be simplified considerably if we work in terms of the deviate process, y t. Noting that the Y t and y t processes are related by y t = Y t µ, (where µ = E(Y t )) we can translate our calculations back into the Y t process simply by adding a constant to our result. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
4 Now there are actually two expected values of a time series process which can be used for univariate forecasts: the unconditional and the conditional process expectations. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
5 Example: ARIMA(1,0,0) To illustrate the differences between these two expectations, consider the ARIMA(1, 0, 0) process y t = φ 1 y t 1 + w t (where φ 1 < 1), this process can be expressed as a sum of past shocks (Remember?) y t = w t + φ 1 w t 1 + φ 2 1w t 2 + φ 3 1w t Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
6 Taking the expected value of this expression, E[y t ] = E[w t ] + φ 1 E[w t 1 ] + φ 2 1E[w t 2 ] + φ 3 1E[w t 3 ] +... and thus E[y t ] = 0 E[Y t ] = E[y t ] + µ = µ. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
7 Extrapolating this term into the future, y t (1) = E[y t+1 ] = 0 y t (2) = E[y t+2 ] = 0... y t (n) = E[y t+n ] = 0 When the unconditional expectation of the process is used as a univariate forecast, the process mean is the forecast regardless of lead time. (The problem with forecasts based on the unconditional expectation of a process is that much valuable information is ignored. ) Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
8 The conditional expectation of y t+1 is: E[y t+1 y t, y t 1,..., y 2, y 1 ]. The conditional expectation of y t+1 is conditional upon the t preceding observations of the time series process. (Recalling that y t+1 = w t+1 + φ 1 y t and y t+1 = w t+1 + φ 1 w t + φ 2 1 w t 1 + φ 3 1 w t 2 + φ 4 1 w t ) E[y t+1 y t, y t 1,..., y 2, y 1 ] = E[w t+1 ] + φ 1 y t = φ 1 y t. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
9 Conditional expectation forecasts of the ARIMA(1,0,0) process are, then, y t (1) = E[w t+1 + φ 1 y t ] = φ 1 y t... y t (n) = E[w t+n + φ 1 w t+n φ n 1 1 w t+1 + φ n 1 y t] = φ 1 y t = φ n 1 y t. It should be intuitively plausible that the best forecast of a time series process is the conditional expectation of the process. What we mean by best in this context is that the conditional expectation forecast has the lowest possible mean-square forecast error (MSFE) of any expectation-based forecast. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
10 Using this conditional expectation as a forecast of y t+1, the error in forecasting is: e t+1 = y t+1 y t (1) = w t+1. This error will always be equal to the random shock, w t+1, and the forecast variance is thus VAR(1) = E[w 2 t+1] = σ 2 w, which is the variance of the white noise process. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
11 A 95% interval forecast of y t+1 is thus y t (1) ± 1.96σ w. (We expect y t+1 to lie in this interval 95% of the time). Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
12 If we now wish to forecast the next value of the process, we begin with the expression for y t+2. y t+2 = w t+2 + φ 1 y t+1 y t+2 = w t+2 + φ 1 w t+1 + φ 2 1y t = w t+2 + φ 1 w t+1 + y t (2). Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
13 The error in forecasting is: e t+2 = y t+2 y t (2) = w t+2 + φ 1 w t+1. The forecast variance is thus VAR(2) = E[(w t+2 + φ 1 w t+1 ) 2 ] = σw 2 (1 + φ 2 1). Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
14 A 95% interval forecast of y t+2 is thus y t (2) ± 1.96 σw 2 (1 + φ 2 1 ). (We expect y t+2 to lie in this interval 95% of the time). Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
15 It can be shown that VAR(1) = σw 2 VAR(2) = (1 + φ 2 1 )σ2 w... VAR(n) = (1 + φ φ2n 2 1 )σw 2. Noting that the expression for VAR(n) is a geometric progression, forecast variance approaches a limit of lim n VAR(n) = 1 φ 2 1 which is the variance of the y t autoregressive process. σ2 w Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
16 Example: Moving averages An ARIMA(0, 0, 1) process written as y t = w t + θ 1 w t 1 Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
17 Point forecasts for the ARIMA(0, 0, 1) are: y t (1) = θ 1 w t y t (2) = 0... y t (n) = 0 Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
18 Forecast variance is given by: VAR(1) = σ 2 w VAR(2) = (1 + θ 2 1 )σ2 w... VAR(n) = (1 + θ 2 1 )σ2 w Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
19 After the second step into the future, forecast variance is constant. The limit of VAR(n) is thus lim n VAR(n) = (1 + θ 2 1)σ 2 w, which is the variance of the ARIMA(0, 0, 1) process. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
20 Example: Integrated Processes An ARIMA(0, 1, 0) process, or random walk, written as y t = y t 1 + w t Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
21 Point forecasts for the ARIMA(0, 1, 0) are: y t (1) = y t y t (2) = y t... y t (n) = y t Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
22 Forecast variance is given by: VAR(1) = σ 2 w VAR(2) = 2σ 2 w... VAR(n) = nσ 2 w Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
23 After two or three steps into the future, the confidence intervals become so large as to render the interval forecast meaningless. Al Nosedal University of Toronto Univariate ARIMA Forecasts (Theory) April 10, / 23
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