Testing for a Change in the Slope Coefficient
Table of Contents
1. Overview
We want to test whether there was a change in the relationship between two time series after some event. Assume we can model the relationship between the series with a linear regression, so we want to test whether the slope coefficient changed after some event. Say we are looking at the relationship between two time-series, \(y\) and \(x\).
A standard regression would be:
\[y_t = \alpha + \beta_1 x_t + \epsilon_t\]
In this case \(\beta_1\) measures the relationship between \(x\) and \(y\) over the entire sample. However say we think the relationship changed at time \(t^*\). We then create an indicator variable (\(I\)) which is 0 before \(t^*\) and 1 after.
Our regression would now be:
\[y_t = \alpha + \beta_1 x_t + \beta_2 (x_t I_t) + \epsilon_t\]
where \(x_t I_t\) is the product of \(x_t\) and the indicator variable at time \(t\).
Note the regression prior to the event is:
\[y_t = \alpha + \beta_1 x_t + \epsilon_t\]
because the indicator is 0. After the event, when the indicator is 1, the equation is:
\[y_t = \alpha + (\beta_1 + \beta_2)x_t + \epsilon_t\]
2. 1 vs 2 Month Crude Oil Futures
library(EIAdata) library(quantmod)
key <- source("~/eia_key")$value one_month <- getEIA('PET.RCLC1.W', key = key) two_month <- getEIA('PET.RCLC2.W', key = key)
2.1. No Change in Slope
\[oil\_2_t = \alpha + \beta oil\_1_t + \epsilon\]
oil_data <- merge.xts(Delt(one_month)[-1], Delt(two_month)[-1,], join="inner") names(oil_data) <- c("one", "two") oil_data <- oil_data['2015-01-01/'] reg1 <- lm(oil_data$two ~ oil_data$one) summary(reg1)
Call:
lm(formula = oil_data$two ~ oil_data$one)
Residuals:
Min 1Q Median 3Q Max
-0.35516 -0.02315 0.00322 0.02237 0.20823
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0007761 0.0021394 0.363 0.717
oil_data$one 0.1298306 0.0135754 9.564 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.04503 on 442 degrees of freedom
Multiple R-squared: 0.1715, Adjusted R-squared: 0.1696
F-statistic: 91.46 on 1 and 442 DF, p-value: < 2.2e-16
2.2. Test for a change in Slope Post 1/1/2020
\[oil\_2_t = \alpha + \beta_1 oil\_1_t + \beta_2 * (oil\_1_t * indicator_t) + \epsilon\]
## create an indicator function----- oil_data$indicator <- ifelse(index(oil_data) < "2020-01-01", 0 , 1) oil_data
one two indicator
2015-01-02 -0.041621816 -0.040627227 0
2015-01-09 -0.087420442 -0.084881129 0
2015-01-16 -0.034256410 -0.033488939 0
2015-01-23 -0.011894647 -0.011969761 0
2015-01-30 -0.017196905 -0.012327311 0
2015-02-06 0.107830271 0.107811491 0
2015-02-13 0.009674235 0.010878011 0
2015-02-20 0.012710207 0.007686395 0
2015-02-27 -0.043637768 -0.019260107 0
2015-03-06 0.017565112 0.015944002 0
...
2023-04-28 -0.036237374 -0.038378993 1
2023-05-05 -0.067732215 -0.066824209 1
2023-05-12 0.012788083 0.013224536 1
2023-05-19 -0.005966422 -0.004582061 1
2023-05-26 0.015494137 0.015204352 1
2023-06-02 -0.039862543 -0.038609508 1
2023-06-09 0.024767359 0.024724882 1
2023-06-16 -0.029896619 -0.029149233 1
2023-06-23 0.014256912 0.015227697 1
2023-06-30 -0.014198495 -0.015282298 1
reg2 <- lm(oil_data$two ~ oil_data$one + I(oil_data$one * oil_data$indicator))
summary(reg2)
Call:
lm(formula = oil_data$two ~ oil_data$one + I(oil_data$one * oil_data$indicator))
Residuals:
Min 1Q Median 3Q Max
-0.256784 -0.002520 -0.000329 0.004423 0.224879
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0004253 0.0017376 0.245 0.807
oil_data$one 0.9598248 0.0559223 17.164 <2e-16 ***
I(oil_data$one * oil_data$indicator) -0.8633876 0.0570306 -15.139 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.03657 on 441 degrees of freedom
Multiple R-squared: 0.4548, Adjusted R-squared: 0.4523
F-statistic: 183.9 on 2 and 441 DF, p-value: < 2.2e-16
Pre covid the slope was 0.96. Post covid the slope was 0.96 + -0.863 = 0.097. The difference in slope coefficients in significant at the 1% level.