ˆ y t +1 = y t + t X i =2 y i - y i - 1 t - 1 for one period in the future. If we extend it further to h periods: ˆ y t + h = y t + h · t X i =2 y i - y i - 1 t - 1 In essence, we are taking the last observation and adding on to it, the average change in values over the entire time frame. 11.4 Model Diagnostic We want to check for autocorrelation to check if the model is capturing patterns adequately. Furthermore, we want to do density forecasting so we need to check if the assumptions we are building for these forecast intervals are appropriate for the models we are estimating. Autocorrelation of time series process is to see the correlation with itself. ⇢ k = E [( Y t - μ )( Y t - k - μ )] σ 2 = Corr ( Y t , Y t - k ) whereby k is the lag, μ σ 2 are the mean and variance of the time series. 55
We can then calculate the sample autocorrelation . ⌧ k = T - k P t =1 ( y t - k - ¯ y )( y t - ¯ y ) T P t =1 ( y - ¯ y ) 2 Corr( Y t , Y t - k ) = Cov ( Y t ,Y t - k ) p Y t p Y t - k We make the assumption that the variance of the time series is constant. From all this, the autocorrelation function (ACF) plot tells us the autocorrelation for a range of lags. White Noise Process Is a sequence of iid random variables ⇠ (0, σ 2 ). We assume that the error terms are white noises. If the model is well specified and captures the time series patterns well, the residuals should behave like a white noise process. Y t = f ( Y 1: t ) + ✏ t ✏ t = f ( Y 1: t ) - Y t so if the error term is autocorrelated, then our model we set up is missing time series pattern and therefore should adjust the model accordingly to get an uncorrelated series of residuals. 3 diagnostic plots we should use: Residual Plot : The presence of patterns in time series of residuals may suggest assumption violations and need for alternative models. Residual ACF Plot : This tells us the autocorrelation for di ff erent lag lengths. If sub- stantial autocorrelation, we have an issue. This is for us to get accurate point forecast. Well specified models should lead to small and insignificant sample autocorrelation, which is consistent with a white noise process. Residual distribution plots : This is useful for density forecast. We can use normal assumption for density forecasting but if not, we need to use bootstrapping and other techniques. This allows us to check the appropriate assumptions for interval forecasting. We can check these via histograms, KDE, Q-Q plots. LOOK AT TUTORIAL ON HOW TO DO ALL OF THESE Time series analysis steps 56
1) Problem definition 2) EDA 3) Modelling 4) Model Diagnostic 5) Model Evaluation 6) Forecast 11.5 Model Validation Recall that model validation looks at whether models we are using are suitable for our needs. We score a given model with a measure of how well it serves its purpose. On the other hand, model selection answers which model is most useful for our needs.
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