use log(y) to forecast y in tsCV function [closed]

Question

I am using tsCV function from Forecast package, I want to have log(y) as the dependent variable and then forecast y rather than forecasting log(y). I tried to obtain the forecast errors from tsCV and then exponentiate them as:

far2_xreg <- function(x, h, xreg, newxreg) { forecast(Arima(x, order=c(0,0,0), xreg=xreg), xreg=newxreg) }

# generating forecast errors
e1 <- mytsCV(log(data$y), far2_xreg, h=7, window = 1000, xreg_actual=xreg_actual, xreg_forecast=xreg_forecast)

data$e1 <- e1

data$e1<−shift(data$e1, n=1L, fill=NA, type=c("lag"), give.names=FALSE)


# exponentiating the forecast erros
data$e<− exp(data$e1)

mape <- mean(abs((data$e)/data$y), na.rm = TRUE)*100

But I know in this form, It is forecasting the log(y) and then it obtaines the errors, and I transform those errors to the original data. However, it seems it is not the correct approach. Can someone guide me about it please, is this possible within this function to use log(y) to forecast y? Thank you very much.

Answer 1

Why would you want to exponentiate the errors? That will not give you anything useful. To see that, think about the equation you are fitting:

$$\log(y_t) = \beta_0 + \beta_1 x_t + \varepsilon_t$$

The errors on the transformed scale are simply the innovation residuals $\hat{\varepsilon}_t$. The errors on the original scale (called response residuals) are $y_t - \hat{y}_t$ where $\hat{y}_t = \exp(\hat\beta_0 + \hat\beta_1 x_t)(1 + \hat\sigma^2/2)$ and $\hat\sigma^2$ is the estimated variance of $\varepsilon_t$. You might use the innovation residuals to check model assumptions (homoscedasticity, normality, autocorrelation, etc). You might use the response residuals if you want to find where the largest forecast errors are, for example.

In any case, the package allows for transformations using the lambda argument (specifying a Box-Cox transformation). In this case, there is no need to explicitly take logs at all. The following code will give innovation residuals.

far2_xreg <- function(x, h, xreg, newxreg) { 
 forecast(Arima(x, order=c(0,0,0), xreg=xreg), xreg=newxreg, lambda=0) 
}

# generating forecast errors
e1 <- mytsCV(data$y, far2_xreg, h=7, window = 1000, xreg_actual=xreg_actual, xreg_forecast=xreg_forecast)