M1<- lm(yºx, data=df2) summary (m1)
Call:
lm (formula = y "x, data = df2)
Residuals:
Min 1Q Median
-65.10 -18.51 -2.10 3Q 17.42 Max 68.55
Coefficients:
Estimate Std. Error t value Pr(>It)
(Intercept) 52.737 1.874 28.14 <2e-16 ***
x 22.923 1.787 12.82 <2e-16 ***
Signif. codes: O *** 0.001 ** 0.01 * 0.05. 0.1 1
Residual standard error: 26.5 on 198 degrees of freedom Multiple R-squared: 0.4537, Adjusted R-squared: 0.451 F-statistic: 164.5 on 1 and 198 DF, p-value: < 2.2e-16
acf (m1$residual)
pacf (m1$residual)
Box. test(m1$residual, lag=10, type="Ljung")
Box-Ljung test
data: m1$residual
X-squared = 177.09, df = 10, p-value < 2.2e-16
m2=arima (y, order = c(1,0,0), xreg = x)
m2
Call:
arima (x = y, order = c(1, 0, 0), xreg = x)
Coefficients:
ar1 intercept x
0.7084 52.6141 24.4899
s. e 0.0496 4.4589 1.0576
sigma 2 estimated as 346.2: log likelihood = -868.84, aic = 1745.68
Box. test(m2$residual, lag=10, type="Ljung")
Box-Ljung test
data: m2$residual
X-squared = 7.534, df = 10, p-value = 0.6743
See the outputs on page 8 after we fit two models for simulated time series data Yt against Xt:
A) Write down the fitted linear regression. What is the R2 of the linear regression? Is the fitted model adequate? Why?
B) After reviewing the ACF and PACF plots. The researcher decided to fit the model 2 with time series errors for this simulated data. What is the fitted model? Write down the fitted model. What is the residual standard deviation of the model? (c) Based on the available output, is model 2 a good fit for the data? Why?