ARIMAX-GARCH flattens when using daily return, but not level

Question

Whenever I do my ARIMAX-GARCH model for forecasting n-ahead with sentiment from news as my exogenous variable, the predictions seems normal when forecasting using level price of the stock, but it heavily underestimates close to 0 when using daily return in percent change or log percent change.

As seen below the curve flattens out, and the results seems weird when I use daily change in %.

Here is my code. I change variable in my zoo when using daily return etc.

library(readxl)
library(vars)
library(tseries)
library(zoo)
library(dplyr)
library(rugarch)
library(forecast)

# Loading Data
finbert_OMX5 <- read_excel("~/desktop/omxc25procent.xlsx")

# Preparing Data
dates <- as.Date(finbert_OMX5$date)
FINsentiment3 <- zoo(finbert_OMX5$finbert_change, dates)
OMXdaily_return3 <- zoo(finbert_OMX5$Price, dates)

# Combined
fin_zoo4 <- merge(FINsentiment3, OMXdaily_return3)
fin_zoo4 <- fin_zoo4[order(index(fin_zoo4)),]

# Split data into train and test
split_index <- round(0.8 * nrow(fin_zoo4))
train_data <- fin_zoo4[1:split_index, ]
test_data <- fin_zoo4[(split_index + 1):nrow(fin_zoo4), ]

train_data_core <- coredata(train_data)
test_data_core <- coredata(test_data)

# Random Walk Model
random_walk_forecasts <- lag(test_data_core[, "OMXdaily_return3"], 1)
random_walk_forecasts[1] <- tail(train_data_core[, "OMXdaily_return3"], 1)  # First element from last of train set

# ARIMAX-GARCH Model Setup and Forecasting
spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
                   mean.model = list(armaOrder = c(2, 2), include.mean = FALSE, external.regressors = matrix(train_data_core[, "FINsentiment3"], ncol = 1)),
                   distribution.model = "norm")
arimax_garch_forecasts <- numeric(nrow(test_data_core))

for(i in 1:nrow(test_data_core)) {
    current_train_data <- fin_zoo4[1:(split_index + i - 1), ]
    current_train_data_core <- coredata(current_train_data)
    
    if(i > 1) {
        xreg_train <- matrix(current_train_data_core[, "FINsentiment3"], ncol = 1)
        xreg_forecast <- matrix(test_data_core[i - 1, "FINsentiment3"], ncol = 1)
        
        spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
                           mean.model = list(armaOrder = c(2, 2), include.mean = FALSE, external.regressors = xreg_train),
                           distribution.model = "norm")
        model <- ugarchfit(spec = spec, data = current_train_data_core[, "OMXdaily_return3"])
        forecast_info <- ugarchforecast(model, xreg = xreg_forecast, n.ahead = 1)
        forecast_mean <- forecast_info@forecast$seriesFor
        arimax_garch_forecasts[i] <- as.numeric(forecast_mean)
    }
}

# Calculate and Compare Metrics for Both Models
calculate_metrics <- function(forecasts, actuals) {
    errors <- forecasts - actuals
    # Exclude the first forecast
    errors <- errors[-1]
    actuals <- actuals[-1]
    list(
        RMSE = sqrt(mean(errors^2)),
        MAPE = mean(abs(errors) / abs(actuals)) * 100,
        SMAPE = mean(2 * abs(errors) / (abs(actuals) + abs(forecasts[-1]))) * 100,
        MedAE = median(abs(errors))
    )
}


arimax_garch_metrics <- calculate_metrics(arimax_garch_forecasts, test_data_core[, "OMXdaily_return3"])
random_walk_metrics <- calculate_metrics(random_walk_forecasts, test_data_core[, "OMXdaily_return3"])

# Print the metrics
cat("ARIMAX-GARCH Model Metrics:\n")
print(arimax_garch_metrics)
cat("\nRandom Walk Model Metrics:\n")
print(random_walk_metrics)

# Comparison table
actual_values <- test_data_core[, "OMXdaily_return3"]
comparison_df <- data.frame(Forecast = arimax_garch_forecasts, RandomWalk = random_walk_forecasts, Actual = actual_values)
print(comparison_df)

# Plotting forecasts and actual values without the first observation
plot_forecasts <- function(actuals, forecasts1, forecasts2, title) {
    plot(actuals[-1], type = "l", col = "blue", ylim = range(c(actuals[-1], forecasts1[-1], forecasts2[-1])),
         main = title, xlab = "Date", ylab = "Price/Returns")
    lines(forecasts1[-1], col = "red")
    lines(forecasts2[-1], col = "green")
    legend("topleft", legend = c("Actual", "ARIMAX-GARCH", "Random Walk"), 
           col = c("blue", "red", "green"), lty = 1)
}

# Plot the forecasts and actual values without the first observation
plot_forecasts(test_data_core[-1, "OMXdaily_return3"], arimax_garch_forecasts[-1], random_walk_forecasts[-1], "ARIMAX-GARCH vs Random Walk Forecast")
```

Answer 1

Your eyes may be tricking you. What matters is the vertical distance between the forecast and the corresponding realization. In the first graph, the vertical distance is quite hard to judge. Also, the scales of the vertical axes are different between the graphs. And the random walk forecast for the returns in the second graph is terrible compared to the ARIMAX-GARCH forecast (again, look at the vertical distance)! All in all, visual comparison between the two graphs is very difficulty and may easily be misleading.