How to Calculate the p-value of Parameters for ARIMA Model in R?

ARIMA (AutoRegressive Integrated Moving Average) is a widely used statistical method for time series forecasting. Evaluating the significance of ARIMA model parameters is essential to understand the model's reliability. The p-value of the parameters helps in determining whether a parameter is significantly different from zero. This article explains how to calculate the p-value of parameters for an ARIMA model in R.

Understanding ARIMA Model Parameters

Three parameters define an ARIMA model:

• p: The number of lag observations included in the model (autoregressive part).
• d: The number of times that the raw observations are differenced (differencing part).
• q: The size of the moving average window (moving average part).

Steps to Calculate the P-value of Parameters for ARIMA Model in R

Here we discuss the main Steps to Calculate the P-value of Parameters for the ARIMA Model in the R Programming Language.

• Load the Data: Import your time series data into R.
• Fit the ARIMA Model: Use the auto.arima or arima function to fit the model.
• Extract the Coefficients: Retrieve the model coefficients.
• Calculate the p-values: Use statistical methods to compute the p-values for the model coefficients.

Step 1: Load the Data

First, let's load some sample time series data.

# Load necessary packages
install.packages("forecast")
library(forecast)

# Load sample time series data
data(AirPassengers)
ts_data <- AirPassengers

Step 2: Fit the ARIMA Model

Fit the ARIMA model using the auto.arima function, which automatically selects the best ARIMA model based on AIC, AICc, or BIC.

# Fit the ARIMA model
fit <- auto.arima(ts_data)
summary(fit)

Output:

Series: ts_data 
ARIMA(2,1,1)(0,1,0)[12] 

Coefficients:
         ar1     ar2      ma1
      0.5960  0.2143  -0.9819
s.e.  0.0888  0.0880   0.0292

sigma^2 = 132.3:  log likelihood = -504.92
AIC=1017.85   AICc=1018.17   BIC=1029.35

Training set error measures:
                 ME     RMSE     MAE      MPE     MAPE     MASE        ACF1
Training set 1.3423 10.84619 7.86754 0.420698 2.800458 0.245628 -0.00124847

Step 3: Extract the Coefficients

Retrieve the coefficients of the fitted ARIMA model.

# Extract coefficients
coefs <- coef(fit)
print(coefs)

Output:

       ar1        ar2        ma1 
 0.5959807  0.2142746 -0.9818772 

Step 4: Calculate the p-values

The p-values are not directly provided by the arima or auto.arima functions. However, you can calculate them using the standard errors of the coefficients. These standard errors are available in the model summary. You can then use these to compute the p-values.

# Extract standard errors
se <- sqrt(diag(vcov(fit)))

# Calculate the t-values
t_values <- coefs / se

# Calculate the p-values
p_values <- 2 * (1 - pnorm(abs(t_values)))

# Create a data frame with coefficients, standard errors, t-values, and p-values
results <- data.frame(
  Coefficients = coefs,
  Std_Errors = se,
  T_Values = t_values,
  P_Values = p_values
)
print(results)

Output:

    Coefficients Std_Errors   T_Values     P_Values
ar1    0.5959807 0.08881790   6.710142 1.944356e-11
ar2    0.2142746 0.08796617   2.435874 1.485584e-02
ma1   -0.9818772 0.02920173 -33.623939 0.000000e+00

• Coefficients: The estimated values of the ARIMA model parameters.
• Standard Errors: The standard errors of the estimated coefficients.
• T-Values: The ratio of the coefficients to their standard errors.
• P-Values: The probability that the observed t-value would occur under the null hypothesis that the true coefficient is zero.
• Significant Coefficients: Parameters with p-values less than a chosen significance level (e.g., 0.05) are considered statistically significant. This suggests that these parameters contribute meaningfully to the model.
• Non-Significant Coefficients: Parameters with p-values greater than the significance level are considered not statistically significant, suggesting they may not contribute significantly to the model.

Conclusion

Calculating the p-values of ARIMA model parameters in R is an important step in time series analysis. It helps in understanding the significance of each parameter, which in turn affects the model's reliability and performance. By following the steps outlined in this guide, you can effectively calculate and interpret the p-values of parameters for an ARIMA model in R.