Unit Root Test with Structural Break
Introduction to Unit Root tests with structural breaks
Welcome to another EViews tutorial. In this article, we will delve into the importance of taking structural breaks into consideration when conducting unit root tests. The conventional unit root tests, such as the Dickey-Fuller test, tend to be inclined towards the non-rejection of a unit root when structural breaks are present, which can lead to erroneous conclusions.
As a result, this tutorial is designed to provide you with a comprehensive understanding of the concept of structural breaks and how to account for them when performing unit root tests in EViews. By the end of this tutorial, you will have gained valuable knowledge on this crucial topic and be better equipped to handle unit root tests with breakpoints in your own research.
NOTE: At the end of the article, you'll find the option to purchase the material needed to replicate the content, as well as a free dataset download. Additionally, a step-by-step video and recommended literature are provided to enhance your understanding of the topic.
Example: Let's Generate an AR(1) Process
To elaborate on this point, let us consider an example. We will examine the case of a discrete change in the mean of an otherwise stationary series. In EViews, we will generate an AR(1) series with a dummy variable for a discrete change in the mean value. The variable "yt" will have the lag value of "yt-1," and we will have a normally distributed error. The gamma here will represent the dummy variable, which will be equal to 4 starting from observation number 50 onwards. Next, we will conduct the Dickey-Fuller test and observe the results
How to Generate an AR(1) process in EViews
We will begin by generating a scalar by typing "scalar a1=0.4". Then, we will generate the error term by typing "gen et," which is the name of the variable, and we will type "nrnd" to generate a random series that is normally distributed.
Now, we generate our series by typing "series y = a1*(y(-1)) + et." This will generate our AR(1) series.
Script
scalar a1=0.4
genr et=nrnd
genr dummy=0
smpl 1 1
genr y=0
smpl 2 100
series y = a1*y(-1)+et
Once the AR(1) process has been generated, the next step is to verify that the series we generated is stationary. We can first examine the graph and then the correlogram.
Upon observation of the graph, it can be inferred that the time series does not exhibit any trend or intercept. Additionally, it can be observed that the series oscillates around a mean value of zero. Moreover, there are no discernible patterns of autocorrelation or heteroskedasticity present in the series.
By examining the correlogram, we can see that this variable is stationary, with no component in the autocorrelation that decays in a slow pattern.
We can even perform the unit root test by putting no intercepts in levels. The null hypothesis is that "y" has a unit root, and if we check the p-value because it is smaller than 0.05, we reject this hypothesis and show that we are working with a stationary variable.
Now, let's see what happens when we include a discrete jump (dummy variable = 4)
Dummy Variable: Discrete jump
When we include the dummy variable in our series, we can observe a discrete jump in what would have been a stationary series. The problem with this is that although the real model would have a behavior like this, EViews will produce inaccurate results when we try to feed a linear regression. Furthermore, if we perform the unit root test (standard unit root test augmented Dickey-Fuller with no intercept or trend), we cannot reject the null hypothesis that "y" has a unit root. This is because the presence of a break, which causes a discrete jump in the mean value, leads our unit root test to suggest that the variable is non-stationary.
Now it becomes apparent that the series appears to be stationary up until the 49th observation, after which a sudden discrete jump occurs. The mean of the series also changes from zero to six after the 49th observation. However, from observation 50 onwards, the series appears to be stationary once again.
What is the problem now?
The issue now is that if we were to fit a linear regression, EViews would fit a line the following way:
In this scenario, the series is deemed non-stationary due to the sudden and significant jump observed at the 50th observation. This jump causes the linear fit to increase over time, further indicating non-stationarity. To better illustrate this, an improved linear fit should resemble the following:
From observation 0 to 49, the series exhibits a consistent, stationary pattern of fluctuations around a mean value of zero. However, after the 49th observation, the series demonstrates a consistent pattern of fluctuations around a mean value of 6.
Augmented Dickey Fuller on the Series with a dummy variable
Upon conducting the augmented Dickey-Fuller test on the series with the discrete jump, it is apparent that the test results indicate non-stationarity. This is demonstrated by the p-value of 0.5849 obtained from the test.
How to tackle a structural break
Option 1: Breaking the sample into two
One possible econometric procedure to address this issue is to break the sample into two, where we suspect the break, and perform the Dickey-Fuller test in each part of the sample. However, the accuracy of the results is subject to selecting the right break date.
To illustrate this, we will divide the sample into two parts, beginning from observation 50 where a discrete jump in the mean value occurred. We will conduct a unit root test on a variable for the first part of the sample until observation 49 where the variable was stationary. Next, we will perform the same test for the second part of the sample, from observation 50 to 100, where the variable was again stationary but with a different intercept due to the observed jump. The test will be conducted using an intercept, which yields a p-value less than 0.05, indicating the rejection of the null hypothesis of the unit root. These results clearly demonstrate that the variable was stationary before the observed break.
Option 2: Perform a Unit Root test with a structural break
Another possibility is conducting a unit root test with a structural break, as Perron did in his 1997 paper, where he challenged Nelson and Plosser's findings in 1982 that most macroeconomic variables have unit root processes. Using the same dataset, Perron's results suggest that these variables are trend-stationary with a structural break. The breakpoint date is selected based on the Dickey Fuller minimum t-statistic, and the lag length is selected using the F-statistic max lag, which is set at 10 by default. We observe a clear break in the intercept in the graph for the real GNP variable in 1929. Therefore, we select the trend and intercept in the unit root test.
The null hypothesis of the unit root is rejected with a p-value less than 0.05, indicating that the variable is stationary with a breakpoint in 1929. The suggested lag length by Perron is eight lags, based on the F-statistic. The selection criteria for the breakpoint date is the minimum t-statistic.
The graph below illustrates the Dickey-Fuller minimum T statistic criterion used to select the structural break date. It is evident that the minimum is reached in 1929, indicating that the test will select 1929 as the break date. These results align with the findings of the original authors of the paper.
Watch the video tutorial and download the dataset
If you have any doubts or questions regarding the steps outlined in this article, we recommend watching the accompanying video tutorial. The tutorial is organized into various chapters, providing viewers with the opportunity to learn and replicate each step discussed in this article.
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Recommended Literature
I would like to recommend some literature to you that I believe will broaden your understanding of macroeconomic time series. These articles explore various aspects of trends, random walks, and unit root hypothesis.
Firstly, I recommend reading "Trends and random walks in macroeconomic time series: Some evidence and implications" by Nelson and Plosser (1982). This article provides empirical evidence and implications for macroeconomic time series, which can help you to understand the patterns of these time series.
Secondly, I recommend reading "The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis" by Pierre Perron (1989). This article focuses on the relationship between the great crash and the oil price shock, and the implications of the unit root hypothesis for macroeconomic time series.
Finally, I recommend reading "further evidence on breaking trend functions in macroeconomic variables" by Pierre Perron (1997). This article presents additional evidence on breaking trend functions in macroeconomic variables, which can provide you with further insights into these time series.
All of these articles are available online, and I encourage you to take the time to read and reflect on them. I hope that they will help you to gain a deeper understanding of macroeconomic time series and the various trends that can be observed in them.