Time Series Analysis - Building ARIMA models

1. Exploratory Data Analysis

The data used in this practice are cow.dat data measuring the daily morning temperature readings for a cow.

Figure 1: Cow Data Plots

Since the data in Figure 1 are not stationary, I decided to used differenced data (d=1).

Figure 2: Differenced Cow Data Plots

Differenced data in Figure 2 seem to be stationary, so I created ACF and PACF plots with differenced data.

Figure 3: ACF and PACF Plots of Differenced Cow Data

Based on the two plots in Figure 3, I set p=2 and q=1. Then, I fitted plots of AR(2), MA(1), and ARMA(2,1) models using sarima() function in R.

2-3. Model Estimation & Model Diagnostics

All data used to fit below models are differenced data (d=1).

a. AR(2) Model

Figure 4: AR(2) Model of Differenced Cow Data

       Estimate       SE    t.value  p.value
ar1     -0.6848   0.1096    -6.2476   0.0000
ar2     -0.3149   0.1104    -2.8537   0.0057
xmean   -0.1514   0.4919    -0.3079   0.7591

In Figure 4, most of p-values are near the blue line. Hence, the H₀ might be wrong, and the residuals might not be white. The estimated coefficients are significant.

b. MA(1) Model

Figure 5: MA(1) Model of Differenced Cow Data

       Estimate       SE    t.value  p.value
ma1     -0.8724   0.0983    -8.8787   0.0000
xmean   -0.2380   0.1339    -1.7766   0.0799

Based on the Figure 5, most of p-values are above the blue line. Hence, the H₀ might not be wrong, and the residuals were white. The estimated coefficients are significant.

c. ARMA(2,1) Model

Figure 6: ARMA(2,1) Model of Differenced Cow Data

      Estimate       SE    t.value   p.value
ar1     0.1368   0.1155     1.1840    0.2404
ar2     0.1866   0.1159     1.6100    0.1119
ma1    -1.0000   0.0555   -18.0304    0.0000
xmean  -0.2396   0.0597    -4.0150    0.0001

Based on the Figure 6, most of p-values are near the blue line. Hence, the H0 might be wrong, and the residuals might not be white. The estimated values of φ's are not significant.

4. Model Selection

(d=1)	AR(2)	MA(1)	ARMA(2,1)
Est Coeffs	All significant	All significant	φ’s not significant
Diagnostics	Ok	Ok	Ok
AIC	7.206017	7.113665	7.127148
AICc	7.21065	7.115949	7.134982
BIC	7.330561	7.207073	7.282828

Table 1: Summary Table

Based on the Table 1, I will choose MA(1) model with differenced data or ARIMA(0,1,1) of the original data.