Question

Please assist with how I can solve this in R

For the usgdp series:

1. if necessary, find a suitable Box-Cox transformation for the data;
2. fit a suitable ARIMA model to the transformed data using auto.arima();
3. try some other plausible models by experimenting with the orders chosen;
4. choose what you think is the best model and check the residual diagnostics;
5. produce forecasts of your fitted model. Do the forecasts look reasonable?
6. compare the results with what you would obtain using ets() (with no transformation).

Answer 1

`Hey,

Note: Brother if you have any queries related the answer please do comment. I would be very happy to resolve all your queries.

# a. if necessary, find a suitable Box-Cox transformation for the data;

autoplot(usgdp)
autoplot(BoxCox(usgdp, BoxCox.lambda(usgdp)))
# When I transformed the original data, I could get more linearly increasing line. Therefore I'm going to do Box-Cox transformation.
lambda_usgdp <- BoxCox.lambda(usgdp)

# b.fit a suitable ARIMA model to the transformed data using auto.arima();

usgdp_autoarima <- auto.arima(usgdp,
lambda = lambda_usgdp)
autoplot(usgdp, series = "Data") +
autolayer(usgdp_autoarima$fitted, series = "Fitted")
# It looked like the model fits well to the data.
usgdp_autoarima
#ARIMA(2, 1, 0) with drift model after Box-Cox transformation.

# c. try some other plausible models by experimenting with the orders chosen;

ndiffs(BoxCox(usgdp, lambda_usgdp))
# the data need 1 first differencing to be stationary.
ggtsdisplay(diff(BoxCox(usgdp, lambda_usgdp)))
# ACF plot shows sinusoidal decrease while PACF plot shows significant spikes at lag 1 and 12. I think that I can ignore the spike at lag 12 because the data are aggregated quarterly, not monthly. Therefore, I'll experiment with ARIMA(1, 1, 0) model.
usgdp_arima.1.1.0 <- Arima(
usgdp, lambda = lambda_usgdp, order = c(1, 1, 0)
)
usgdp_arima.1.1.0
autoplot(usgdp, series = "Data") +
autolayer(usgdp_arima.1.1.0$fitted, series = "Fitted")
# I'll also try ARIMA(1, 1, 0) with drift model.
usgdp_arima.1.1.0.drift <- Arima(
usgdp, lambda = lambda_usgdp, order = c(1, 1, 0),
include.drift = TRUE
)
usgdp_arima.1.1.0.drift
autoplot(usgdp, series = "Data") +
autolayer(usgdp_arima.1.1.0.drift$fitted, series = "Fitted")
# It looked like that these models also fit well to the data.

# d. choose what you think is the best model and check the residual diagnostics;

accuracy(usgdp_autoarima)
accuracy(usgdp_arima.1.1.0)
accuracy(usgdp_arima.1.1.0.drift)
# Some errors show that ARIMA(2, 1, 0) with drift is the best model while others show that ARIMA(1, 1, 0) with drift is the best. Check the residuals of both cases.
checkresiduals(usgdp_autoarima)
checkresiduals(usgdp_arima.1.1.0.drift)
# In either case, the residuals are like white noise series and are not normally distributed.
# I'll choose the best model as ARIMA(2, 1, 0) with drift model. With the model, RMSE and MASE values were lower. And there wasn't significant spike at lag 2 in ACF plot of ARIMA(2, 1, 0) with drift model, even if it exists in ARIMA(1, 1, 0) with drift model.

# e. produce forecasts of your fitted model. Do the forecasts look reasonable?

fc_usgdp_autoarima <- forecast(
usgdp_autoarima
)
autoplot(fc_usgdp_autoarima)
# It looked like the forecasts are reasonable.

# f. compare the results with what you would obtain using ets() (with no transformation).

fc_usgdp_ets <- forecast(
ets(usgdp)
)
autoplot(fc_usgdp_ets)
# It looked like these forecasts are more likely than the ones with ARIMA model. When trend is obvious, is ETS better than ARIMA model? I wonder about it.
```

Kindly revert for any queries

Thanks.