Dynamic harmonic regression (dhr) models

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Jun 16, 2022 01:21 PM

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Applied forecasting

Economics

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We have learnt ARIMA and regression previously, but both have their advantage and disadvantage.

For ARIMA, it is univariate which means that it can be only performed on a single time series. So, say, the effects of holidays, competitor activity, changes in the law, the wider economy, or other external variables, may explain some of the historical variations and may lead to more accurate forecasts.

For regression, allows for the inclusion of a lot of relevant information from predictor variables, but does not allow for the subtle time series dynamics that can be handled with ARIMA models.

Solution : ARIMA models with allowing other information to be included - ARIMAX

Regression with ARIMA errors

👹

Regression model (Time Series) (RTS). we learnt this formula where we assumed to be WN. Now, we want the to be autocorrelated.

You always write it in two equations:

The first equation is the regression part so that part looks the same; it is just that the epsilon is replaced by a eta. The second part is a separate ARIMA model for

So there is still a WN in our model, but in the second part.

Estimation

Plotting the residual of the error


fit <- us_change %>%
  model(
    ARIMA(Consumption ~ Income)
  )
gg_tsresiduals(fit) ## this one
================================== OR this one
residuals(fit, type = "innovation") %>%
  gg_tsdisplay(.resid, plot_type = "partial") +
  ggtitle("ARIMA errors")

Plotting the residual of the error


residuals(fit, type = "regression") %>%
  gg_tsdisplay(.resid, plot_type = "partial") +
  ggtitle("Regression errors")

Forecasting using dynamic regression


## predict 11 periods ahead
us_change_future <- new_data(us_change, 11) %>%
  mutate(
## take the max of income as a scenario
    Income = max(us_change$Income), 
    Savings = mean(us_change$Savings),
    Unemployment = mean(us_change$Unemployment)
  )
forecast(fit, new_data = us_change_future) %>%
  autoplot(us_change) +
  labs(
    x = "Year", y = "Percentage change",
    title = "Forecasts from regression with ARIMA(4,0,4) errors"
  )

Solution : ARIMA models with allowing other information to be included - ARIMAX

Regression with ARIMA errors

Estimation

Forecasting using dynamic regression

In summary, ARIMAX is

Stochastic & deterministic trends

Dynamic harmonic regression

Lagged predictors

FAQ

Math