ECO 5316 - Homework 4
Bobby Merriman
February 18, 2017
Introduction
The purpose of this assignment is to employ the ADF, KPSS, and ERS tests on six different categories of data. We start by giving an introduction as to why these tests are important, and lay a foundation of the intuition behind these tests in section I. In section II we describe the data and how we manipulate the data. Section II gives us a preliminary run down (an informal argument) of why we think we should employ these tests. Section III then runs the ADF test on the data. Section IV then runs the KPSS test. Section V runs the ERS test. Section VI runs all three tests on the first differences of each category of data. Section VII runs the second difference test on one of our data that failed in Section VI. Section VIII concludes and provides the summary of our results.
Section I - Background
Stationarity is an important feature of a time series data set ost of our forecating techniques are based on the assumption of stationarity. In other words, to forecast a time series properly we need the time series to be stationary. There are varying levels of stationarity the most common being strong (strict) and weakly stationary. A strong or strictly stationary time series is a time series in which the joint distribution of the time series data is constant over time; that is, the model that produces the data we are interested in does not change over time. Hence, for a strictly stationary times series, all of the parameters will be constant and thus we should expect the estimates of these parameters (statistics such as mean, variance, etc.) to be statistically indifferent from one another. Weakly stationary occurs when the mean and covariance of the time series data is time invariant; that is consant, does not change over time. Obvisouly in practice we will see more weakly stationary time series, and strictly stationary time series are more of a theoretical idea than an a real process we observe.
However, as one might suspect, most time series data will not be stationary,that is, non-stationary. Non-stationarity can be “fixed” by transforming the data in such a way that the resulting transformed data is approximately weakly stationary. This process of transformation is useful since it then allows us to forecast non-stationary time series. Effectively we perform the forecast on the transformed data (which is stationary) and then “untransform” the forecast (using the inverse of the transformation process) to see the forecast of our original data (the non-stationary process).
Three common forms of non-stationary time series are:
- Trend Stationary
- Difference Stationary with no drift
- Difference Stationary with drift
A trend stationary time series is a function of a non-constant but deterministic mean and a stationary stochastic process with mean zero. The mean is non constant (function of time) and thus the time series is not weakly-stationary. To make a trend stationary time series weakly stationary we estimate the mean trend, subtract this from our time series data, which produces residuals. If the time series is truly trend stationary then the residuals (after correcting for the mean trend) will be a stationary process (by definition trend stationarity).
A difference stationary time series is a function of a lagged component of the dependent varaible and some stochastic portion. The variance is non-constant (function of time) and therefore the time series is not weakly stationary. To correct this non-stationarity we take the difference of the dependent variable with its lag; this yields a stationary stochastic process.
A difference stationary time series can also have a drift term. This type of time series is the same as a difference stationary time series except that the mean is also no longer constant. In other words, a difference stationary time series with drift has neither constant mean nor constant variance and thus is not weakly stationary. To correct this we can generally take the log of the time series and then difference between the log and one of its lags.
The purpose of this paper is to test whether or not our data is trend or difference stationary. We then take the first or second difference of our data to determine if these transformations are “good enough” to say that our transformed data is at least approximately stationary.
Section II - The Data and Manipulation
For this assignment we will look at five categories of data:
- Category A: Real Personal Consumption Expenditures per Capita
- Category B: S&P 500 Index
- Category C: Gross Domestic Product: Implicit Price Deflator
- Category D: 10-Year Treasury Constant Maturity Rate
- Category E: Unemployment Rate
- Category F: U.S. / U.K. Foreign Exchange Rate
Below is the code that we use to plot a time series of each of the different categories of data.
for(i in 1:length(names.of.data)){
data <- eval(parse(text = paste("data.", letters[i], sep = "")))
plot(data, main = paste("Time Series of", names.of.data[i]), xlab = "Year", ylab = names.of.data[i])
rm(data)
}Following this we also plot the time series of the log transformations of each category of data using the following code.
for(i in 1:length(names.of.data)){
log.data <- eval(parse(text = paste("log.data.", letters[i], sep = "")))
plot(log.data, main = paste("Time Series of the Log of", names.of.data[i]), xlab = "Year", ylab = paste("Log of", names.of.data[i]))
rm(log.data)
}Using the first two plots we determine whether or not the data needs to be log-transformed. Finally we plot the first differences of either the orginal data, or the log-transformed data, which will be indicated in the plot title. The code to produce these graphs is given below.
for(i in 1:length(names.of.data)){
if(log.transform.dummy[i] == 0){ diff.data <- eval(parse(text = paste("diff.data.", letters[i], sep = "")))
plot(diff.data, xlab = "Year", ylab = names.of.data[i],
main = paste("Time Series of First Differences of", names.of.data[i]))
rm(diff.data)}
if(log.transform.dummy[i] == 1){ diff.log.data <- eval(parse(text = paste("log.diff.data.", letters[i], sep = "")))
plot(diff.log.data, xlab = "Year", ylab = paste("Log of", names.of.data[i]),
main = paste("Time Series of of First Differences of the Log of", names.of.data[i]))
rm(diff.log.data)}
}For convience sake we now show all three plots for each category in one figure.
## Category A - Real Personal Consumption Expenditures per Capita
Figure 1
## Category B - S&P 500 Index
Figure 2
## Category C - GDP: Implicit Price Deflator
Figure 3
## Category D - 10-year Treasury Constant Maturity Rate
Figure 4
## Category E - Unemployment Rate
Figure 5
## Category F - U.S. / U.K. Foreign Exchange Rate
Figure 6
The graphs are important as they allow us to visually see that each category of data might not be stationary. Recall that stationary implies a constant mean and variance over time. Hence, if the data has a constant mean we might expect that the data fluctuate around the mean over time; in other words, we should be able to draw a flat, horizontal line on the plots that are stationary. This appears to only be true for the first difference data. Hence, we might expect that in order for us to assume stationarity on any category of our data we will need to transform the data by taking the first difference.
A more formal way to test if our data need to be transformed using first (or second) differences, is by performing the ADF, KPSS, and ERS tests on our data before and after we take the first difference. All of these tests will mathematically indicate whether or not statistically our data are produced by a trend or difference stationary model. We now note that we will use the log-transformed data for all of our remaining tests.
Section III - Augmented Dickey-Fuller Test (ADF)
The first test we run is an ADF test. The null hypothesis of this test is that the time series has a unit root. A unit root implies that the time series will be difference stationary, that is, the mean is non-constant, non-zero, and is stochastic. The alternative hypothesis is that the time series is stationary, level stationary, or trend stationary depending upon which specification you use in the test. A stationary time series under this test represents a time series with a constant mean of zero. A level stationary time series is a time series with a non-zero but constant mean. A trend stationary time series has a mean that is non-constant, and non-zero, but is deterministic. In short, this test assumes the time series is difference stationary and tests to see if there is enough evidence to reject this hypothesis and conclude that the time series is trend stationary. The code we use to run these tests is given below:
for(i in 1:length(names.of.data)){
# ADF on Original Data
if(log.transform.dummy[i] == 0){ ADF.data.name <- paste("ADF.", letters[i], sep = "")
ADF.values <- summary(
ur.df(eval(parse(text = paste("data.", letters[i], sep = ""))),
type = "trend", selectlags = "BIC"))
assign(ADF.data.name, ADF.values)
rm(ADF.data.name)
rm(ADF.values)}
# ADF on Log-Transformed Data
if(log.transform.dummy[i] == 1){ ADF.data.name <- paste("ADF.", letters[i], sep = "")
ADF.values <- summary(
ur.df(eval(parse(text = paste("log.data.", letters[i], sep = ""))),
type = "trend", selectlags = "BIC"))
assign(ADF.data.name, ADF.values)
rm(ADF.data.name)
rm(ADF.values)}
}For the sake of space we do not include the test results here but show them in Appendix A. In this section we only suggest the conclusions of the ADF test on each category of data. Note: although there are three tests using the ADF method, we will only focus on the alternative hypothesis which suggests the time series is trend stationary (as opposed to stationary, or level stationary).
ADF Results - Real Personal Consumption Expenditures per Capita
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the Real Personal Consumption Expenditures per Capita time series is not trend stationary. Therefore, the ADF test suggests that the Real Personal Consumption Expenditures per Capita time series is difference stationary.
ADF Results - S&P 500 Index
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the S&P 500 Index time series is not trend stationary. Therefore, the ADF test suggests that the S&P 500 Index time series is difference stationary.
ADF Results - Gross Domestic Product: Implicit Price Deflator
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the Gross Domestic Product: Implicit Price Deflator time series is not trend stationary. Therefore, the ADF test suggests that the Gross Domestic Product: Implicit Price Deflator is difference stationary.
ADF Results - 10-year Treasury Constant Maturity Rate
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the 10-year Treasury Constant Maturity Rate time series is not trend stationary. Therefore, the ADF test suggests that the 10-year Treasury Constant Maturity Rate is difference stationary.
ADF Results - Unemployment Rate
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the Unemployment Rate time series is not trend stationary. Therefore, the ADF test suggests that the Unemployment Rate is difference stationary.
ADF Results - U.S. / U.K. Foreign Exchange Rate
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the U.S. / U.K. Foreign Exchange Rate time series is not trend stationary. Therefore, the ADF test suggests that the U.S. / U.K. Foreign Exchange Rate is difference stationary.
Section IV - Kwiattkowski-Phillipps-Schmidt-Shin Test (KPSS)
The second test we run is a KPSS test. The null hypothesis of this test is that the time series is stationary (mean or trend stationary). Recall, that a trend stationary time series has a mean that is non-constant, and non-zero, but is deterministic. The alternative hypothesis is that the time series is difference stationary (has a unit root). In short, this test assumes that the time series is trend stationary and tests to see if there is enough evidence to reject this hypothesis and conclude that the time series is difference stationary. The code we use to run these tests is given below:
for(i in 1:length(names.of.data)){
# KPSS on Original Data
if(log.transform.dummy[i] == 0){ KPSS.data.name <- paste("KPSS.", letters[i], sep = "")
KPSS.values <- summary(
ur.kpss(eval(parse(text = paste("data.", letters[i], sep = ""))),
type = "tau", lags = "short"))
assign(KPSS.data.name, KPSS.values)
rm(KPSS.data.name)
rm(KPSS.values) }
# KPSS on Log-Transformed Data
if(log.transform.dummy[i] == 1){ KPSS.data.name <- paste("KPSS.", letters[i], sep = "")
KPSS.values <- summary(
ur.kpss(eval(parse(text = paste("log.data.", letters[i], sep = ""))),
type = "tau", lags = "short"))
assign(KPSS.data.name, KPSS.values)
rm(KPSS.data.name)
rm(KPSS.values)}
}For the sake of space we do not include the test results here but show them in Appendix B. In this section we only suggest the conclusions of the KPSS test on each category of data. It is also important to note that we only perform a KPSS test using the tau statistic. The tau statistic assumes that the model is trend stationary (mean is constant but with a linear trend), where as the mu statistic assumes that the model if mean stationary, that is, the mean is constant (determinisitc) and non-zero.
KPSS Results - Real Personal Consumption Expenditures per Capita
The test-statistic for the tau variable is much larger than the critical value even at the 1% level. Hence we can reject the null hypothesis that the Real Personal Consumption Expenditures per Capita time series is trend stationary. In other words, the KPSS test suggests that the Real Personal Consumption Expenditures per Capita time series is difference stationary. It is important to note that these results are consistent with the ADF test (which is not always the case).
KPSS Results - S&P 500 Index
The test-statistic for the tau variable is much larger than the critical value even at the 1% level. Hence we can reject the null hypothesis that the S&P 500 Index time series is trend stationary. In other words, the KPSS test suggests that the S&P 500 Index time series is difference stationary. It is important to note that these results are consistent with the ADF test (which is not always the case).
KPSS Results - Gross Domestic Product: Implicit Price Deflator
The test-statistic for the tau variable is much larger than the critical value even at the 1% level. Hence we can reject the null hypothesis that the Gross Domestic Product: Implicit Price Deflator time series is trend stationary. In other words, the KPSS test suggests that the Gross Domestic Product: Implicit Price Deflator time series is difference stationary. It is important to note that these results are consistent with the ADF test (which is not always the case).
KPSS Results - 10-year Treasury Constant Maturity Rate
The test-statistic for the tau variable is much larger than the critical value even at the 1% level. Hence we can reject the null hypothesis that the 10-year Treasury Constant Maturity Rate time series is trend stationary. In other words, the KPSS test suggests that the 10-year Treasury Constant Maturity Rate time series is difference stationary. It is important to note that these results are consistent with the ADF test (which is not always the case).
KPSS Results - Unemployment Rate
The test-statistic for the tau variable is much larger than the critical value even at the 1% level. Hence we can reject the null hypothesis that the Unemployment Rate time series is trend stationary. In other words, the KPSS test suggests that the Unemployment Rate time series is difference stationary. It is important to note that these results are consistent with the ADF test (which is not always the case).
KPSS Results - U.S. / U.K. Foreign Exchange Rate
The test-statistic for the tau variable is much larger than the critical value even at the 1% level. Hence we can reject the null hypothesis that the U.S. / U.K. Foreign Exchange Rate time series is trend stationary. In other words, the KPSS test suggests that the U.S. / U.K. Foreign Exchange Rate time series is difference stationary. It is important to note that these results are consistent with the ADF test (which is not always the case).
Section V - Elliot, Rothenberg, and Stock Test (ERS)
The third and final test we run is an ERS test. The ERS is a similar test to that of the ADF test we used earlier, except that this test is more powerful. Hence, the null hypothesis is that the time series is difference stationary and the alternative hypothesis is that the time series is trend stationary. The code we use to run these tests is given below:
for(i in 1:length(names.of.data)){
# ERS on Original Data
if(log.transform.dummy[i] == 0){ ERS.data.name <- paste("ERS.", letters[i], sep = "")
ERS.values <- summary(
ur.ers(eval(parse(text = paste("data.", letters[i], sep = ""))),
type = "DF-GLS", model = "trend"))
assign(ERS.data.name, ERS.values)
rm(ERS.data.name)
rm(ERS.values)}
# ERS on Log-Transformed Data
if(log.transform.dummy[i] == 1){ ERS.data.name <- paste("ERS.", letters[i], sep = "")
ERS.values <- summary(
ur.ers(eval(parse(text = paste("log.data.", letters[i], sep = ""))),
type = "DF-GLS", model = "trend"))
assign(ERS.data.name, ERS.values)
rm(ERS.data.name)
rm(ERS.values)}
}For the sake of space we do not include the test results here but show them in Appendix C. In this section we only suggest the conclusions of the ERS test on each category of data. It is also important to note that we only perform a DF-GLS test and not a P-test. This is because an ERS P-test is a more powerful version of the Phillips-Perron (PP) test, which we do not look at in this paper. However, an ERS DF-GLS test is a more powerful version of the ADF test which we have used previously.
ERS Results - Real Personal Consumption Expenditures per Capita
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the Real Personal Consumption Expenditures per Capita time series is not trend stationary. Therefore, the ERS test suggests that the Real Personal Consumption Expenditures per Capita time series is difference stationary. Since this test is more powerful than ADF and the results of both the ADF and ERS are consistent with the KPSS test then we are confident in these results.
ERS Results - S&P 500 Index
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the S&P 500 Index time series is not trend stationary. Therefore, the ERS test suggests that the S&P 500 Index time series is difference stationary. Since this test is more powerful than ADF and the results of both the ADF and ERS are consistent with the KPSS test then we are confident in these results.
ERS Results - Gross Domestic Product: Implicit Price Deflator
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the Gross Domestic Product: Implicit Price Deflator time series is not trend stationary. Therefore, the ERS test suggests that the Gross Domestic Product: Implicit Price Deflator is difference stationary. Since this test is more powerful than ADF and the results of both the ADF and ERS are consistent with the KPSS test then we are confident in these results.
ERS Results - 10-year Treasury Constant Maturity Rate
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the 10-year Treasury Constant Maturity Rate time series is not trend stationary. Therefore, the ERS test suggests that the 10-year Treasury Constant Maturity Rate is difference stationary. Since this test is more powerful than ADF and the results of both the ADF and ERS are consistent with the KPSS test then we are confident in these results.
ERS Results - Unemployment Rate
The test-statistic for the tau variable is large enough to reject the null hypothesis at the 5% level. From this test it appears that the unemployment rate may be trend stationary. However, earlier, both the ADF and KPSS test both suggested that the Unemployment Rate time series was not trend stationary. Hence, we have conflicting results especially since the ERS test is a more powerful version of the ADF test.
ERS Results - U.S. / U.K. Foreign Exchange Rate
The test-statistic for the tau variable is not large enough to reject the null hypothesis even at the 10% level. From this test we are fairly confident that the U.S. / U.K. Foreign Exchange Rate time series is not trend stationary. Therefore, the ERS test suggests that the U.S. / U.K. Foreign Exchange Rate is difference stationary.
Section VI - ADF, KPSS, and ERS tests on 1st Differences
The previous three sections using the ADF, KPSS, and ERS test on the log-transformed data shows us that there seems to be a unit root in each category of our data. Hence, we will now take the first difference of the log-transformed data and rerun all of our tests. The code to run these tests is not given below for the sake of space, and we again point out that the tests are the exact same except that we change the data set that we input into the tests. In other words the code is the exact same as above except for a name change in the data set that we are using; this is why we do not include the code. The resultsare given in Appendic D, E, and F. The conclusions from these tests are given below.
Test Results on First Differences of Real Personal Consumption Expenditures per Capita
The ADF test: The test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of Real Personal Consumption Expenditures per Capita has a unit root. In other words this test suggests that the Real Personal Consumption Expenditures per Capita is difference stationary since the ADF test on the Real Personal Consumption Expenditures per Capita failed to reject the null, but the ADF test on the first difference of Real Personal Consumption Expenditures per Capita easily rejects the null.
The KPSS test: The test-statistic for the tau varaiable is smaller than the critical values even at the 10% level. Hence, we are not able to reject the null that the first difference of Real Personal Consumption Expenditures per Capita is stationary. In other words, this test suggests that the first difference transformation of the log of Real Personal Consumption Expenditures per Capita is enough to make our time series at least weakly stationary.
The ERS test: Similarly to the ADF test, we see that the test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of Real Personal Consumption Expenditures per Capita has a unit root. In other words this test suggests that the Real Personal Consumption Expenditures per Capita is difference stationary since the ADF test on the Real Personal Consumption Expenditures per Capita failed to reject the null, but the ADF test on the first difference of Real Personal Consumption Expenditures per Capita easily rejects the null.
Test Results on First Differences of S&P 500 Index
The ADF test: The test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of S&P 500 Index has a unit root. In other words this test suggests that the S&P 500 Index is difference stationary since the ADF test on the S&P 500 Index failed to reject the null, but the ADF test on the first difference of S&P 500 Index easily rejects the null.
The KPSS test: The test-statistic for the tau varaiable is smaller than the critical values even at the 10% level. Hence, we are not able to reject the null that the first difference of S&P 500 Index is stationary. In other words, this test suggests that the first difference transformation of the log of S&P 500 Index is enough to make our time series at least weakly stationary.
The ERS test: Similarly to the ADF test, we see that the test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of S&P 500 Index has a unit root. In other words this test suggests that the S&P 500 Index is difference stationary since the ADF test on the S&P 500 Index failed to reject the null, but the ADF test on the first difference of S&P 500 Index easily rejects the null.
Test Results on First Differences of Gross Domestic Product: Implicit Price Deflator
The ADF test: The test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of Gross Domestic Product: Implicit Price Deflator has a unit root. In other words this test suggests that the Gross Domestic Product: Implicit Price Deflator is difference stationary since the ADF test on the S&P 500 Index failed to reject the null, but the ADF test on the first difference of Gross Domestic Product: Implicit Price Deflator Index easily rejects the null.
The KPSS test: The test-statistic for the tau varaiable is larger than the critical values even at the 1% level. Hence, we are able to reject the null that the first difference of Gross Domestic Product: Implicit Price Deflator is not stationary. In other words, this test seemingly presents conflicting results to the previous ADF test. We now run a more powerful version of this ADF test, namely the ERS test before proceeding.
The ERS test: Similarly to the ADF test, we see that the test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of Gross Domestic Product: Implicit Price Deflator has a unit root. In other words this test suggests that the Gross Domestic Product: Implicit Price Deflator is difference stationary since the ADF test on the Gross Domestic Product: Implicit Price Deflator failed to reject the null, but the ADF test on the first difference of Gross Domestic Product: Implicit Price Deflator easily rejects the null.
Since we have conflicting evidence we will take the second difference of the Gross Domestic Product: Implicit Price Deflator data and rerun the above tests shown in Section VI.
Test Results on First Differences of 10-year Treasury Constant Maturity Rate
The ADF test: The test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of 10-year Treasury Constant Maturity Rate has a unit root. In other words this test suggests that the 10-year Treasury Constant Maturity Rate is difference stationary since the ADF test on the 10-year Treasury Constant Maturity Rate failed to reject the null, but the ADF test on the first difference of 10-year Treasury Constant Maturity Rate easily rejects the null.
The KPSS test: The test-statistic for the tau varaiable is smaller than the critical values even at the 10% level. Hence, we are not able to reject the null that the first difference of 10-year Treasury Constant Maturity Rate is stationary. In other words, this test suggests that the first difference transformation of the log of 10-year Treasury Constant Maturity Rate is enough to make our time series at least weakly stationary.
The ERS test: Similarly to the ADF test, we see that the test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of 10-year Treasury Constant Maturity Rate has a unit root. In other words this test suggests that the 10-year Treasury Constant Maturity Rate is difference stationary since the ADF test on the 10-year Treasury Constant Maturity Rate failed to reject the null, but the ADF test on the first difference of 10-year Treasury Constant Maturity Rate easily rejects the null.
Test Results on First Differences of Unemployment Rate
The ADF test: The test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of Unemployment Rate has a unit root. In other words this test suggests that the Unemployment Rate is difference stationary since the ADF test on the Unemployment Rate failed to reject the null, but the ADF test on the first difference of Unemployment Rate easily rejects the null.
The KPSS test: The test-statistic for the tau varaiable is smaller than the critical values even at the 10% level. Hence, we are not able to reject the null that the first difference of Unemployment Rate is stationary. In other words, this test suggests that the first difference transformation of the log of Unemployment Rate is enough to make our time series at least weakly stationary.
The ERS test: Similarly to the ADF test, we see that the test-statistic for the tau variable is larger than the critical value at the 5% level. We do point out that this test fails to reject the null at the 1% level. However, this should not concern us since the ADF rejected at the 1% level (and the ERS is simply a more powerful version than the ADF). Furthermore the KPSS rejected the null at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of Unemployment Rate has a unit root. In other words this test suggests that the Unemployment Rate is difference stationary since the ADF test on the Unemployment Rate failed to reject the null, but the ADF test on the first difference of Unemployment Rate easily rejects the null.
Test Results on First Differences of U.S. / U.K. Foreign Exchange Rate
The ADF test: The test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of U.S. / U.K. Foreign Exchange Rate has a unit root. In other words this test suggests that the U.S. / U.K. Foreign Exchange Rate is difference stationary since the ADF test on the U.S. / U.K. Foreign Exchange Rate failed to reject the null, but the ADF test on the first difference of U.S. / U.K. Foreign Exchange Rate easily rejects the null.
The KPSS test: The test-statistic for the tau varaiable is smaller than the critical values even at the 10% level. Hence, we are not able to reject the null that the first difference of U.S. / U.K. Foreign Exchange Rate is stationary. In other words, this test suggests that the first difference transformation of the log of U.S. / U.K. Foreign Exchange Rate is enough to make our time series at least weakly stationary.
The ERS test: Similarly to the ADF test, we see that the test-statistic for the tau variable is much larger than the critical value at the 1% level. Therefore, we are able to reject the null hypothesis that the first difference of U.S. / U.K. Foreign Exchange Rate has a unit root. In other words this test suggests that the U.S. / U.K. Foreign Exchange Rate is difference stationary since the ADF test on the U.S. / U.K. Foreign Exchange Rate failed to reject the null, but the ADF test on the first difference of U.S. / U.K. Foreign Exchange Rate easily rejects the null.
Section VII - ADF, KPSS, and ERS tests on 2nd Differences
From the previous section there was only one category of data on which the tests seemed to present conflicting evidence, namely, the Gross Domestic Product: Implicit Price Deflator time series data. Hence, in this section we re-run the tests that we performed in the previous two sections on the second differences of the Gross Domestic Product: Implicit Price Deflator time series.
We do not show the code here but we do show the code in Appendix G. The reason we show the code in the appendix is because we have generalized our code to work on any D differences in any of our data sets, so that the code can easily be re run for any change in the data or any new data set. We show the results of our tests on the second difference in the Gross Domestic Product: Implicit Price Deflator time series in Appendix H, but we comment on the results below.
Like before the ADF tau statistic is much larger than the critical value at the 1% level. Hence we can reject the null that the second differences of the Gross Domestic Product: Implicit Price Deflator have a unit root, and thus the second differences are approximately stationary according to this test. Unlike before, the KPSS test tau statistic is much smaller than the critical values even at the 10% level. Hence, we fail to reject the null which says that the second difference in Gross Domestic Product: Implicit Price Deflator is stationary. Finally our ERS test, a more powerful test of the ADF test, gives us the same results as the ADF test. Therefore, we can say that the Gross Domestic Product: Implicit Price Deflator time series is approximately stationary when we transform the data by taking the log and then the second difference.
Section VIII - Conclusion
In this paper we took six categories of data and tested whether or not the data had a unit root. In short, we found that all of our data was difference stationary. We found that all the data except for one is approximately stationary when we take the first difference, and that one of our data sets is approximately stationary when we take the second difference. The results are summaried in the table below.
| Category of Data | Type of Time Series Model |
|---|---|
| Real Personal Consumption Expenditures per Capita | I1 |
| S&P 500 Index | I1 |
| GDP: Implicit Price Deflator | I2 |
| 10-year Treasury Constant Maturity Rate | I1 |
| Unemployment Rate | I1 |
| U.S. / U.K. Foreign Exchange Rate | I1 |
Appendices
Appendix A - ADF Test Results
Below are the results for the Real Personal Consumption Expenditures per Capita time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.044251 -0.003656 0.000096 0.004137 0.039159
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.095e-01 1.018e-01 1.075 0.283
## z.lag.1 -1.156e-02 1.131e-02 -1.022 0.308
## tt 6.042e-05 6.420e-05 0.941 0.347
## z.diff.lag 9.595e-02 6.060e-02 1.583 0.114
##
## Residual standard error: 0.008129 on 274 degrees of freedom
## Multiple R-squared: 0.01424, Adjusted R-squared: 0.003448
## F-statistic: 1.319 on 3 and 274 DF, p-value: 0.2683
##
##
## Value of test-statistic is: -1.0222 22.7077 0.8475
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.98 -3.42 -3.13
## phi2 6.15 4.71 4.05
## phi3 8.34 6.30 5.36Below are the results for the S&P 500 Index time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.227851 -0.004385 0.000173 0.004672 0.109422
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.294e-03 8.332e-04 2.754 0.005901 **
## z.lag.1 -6.338e-04 2.642e-04 -2.399 0.016437 *
## tt 1.686e-07 7.400e-08 2.279 0.022709 *
## z.diff.lag 2.764e-02 7.692e-03 3.593 0.000328 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.009691 on 16886 degrees of freedom
## Multiple R-squared: 0.001095, Adjusted R-squared: 0.0009175
## F-statistic: 6.17 on 3 and 16886 DF, p-value: 0.0003468
##
##
## Value of test-statistic is: -2.3993 6.7998 2.9341
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Below are the results for the Gross Domestic Product: Implicit Price Deflator time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0225778 -0.0016538 -0.0002037 0.0015404 0.0190168
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.985e-03 4.471e-03 0.891 0.374
## z.lag.1 -7.417e-04 1.830e-03 -0.405 0.686
## tt 4.004e-06 1.676e-05 0.239 0.811
## z.diff.lag 7.672e-01 3.916e-02 19.591 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.003994 on 274 degrees of freedom
## Multiple R-squared: 0.597, Adjusted R-squared: 0.5926
## F-statistic: 135.3 on 3 and 274 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -0.4052 7.3786 0.4741
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.98 -3.42 -3.13
## phi2 6.15 4.71 4.05
## phi3 8.34 6.30 5.36Below are the results for the 10-year Treasury Constant Maturity Rate time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.185404 -0.004964 -0.000134 0.004810 0.095428
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.483e-03 6.105e-04 2.430 0.01512 *
## z.lag.1 -5.820e-04 2.617e-04 -2.224 0.02618 *
## tt -7.416e-08 3.214e-08 -2.307 0.02105 *
## z.diff.lag 2.778e-02 8.521e-03 3.261 0.00111 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0127 on 13761 degrees of freedom
## Multiple R-squared: 0.001253, Adjusted R-squared: 0.001036
## F-statistic: 5.756 on 3 and 13761 DF, p-value: 0.0006251
##
##
## Value of test-statistic is: -2.2237 2.2726 3.356
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Below are the results for the Unemployment Rate time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.23080 -0.02071 -0.00072 0.02016 0.22968
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.936e-02 8.135e-03 2.380 0.0175 *
## z.lag.1 -1.139e-02 5.013e-03 -2.271 0.0234 *
## tt 1.116e-06 5.927e-06 0.188 0.8507
## z.diff.lag 1.356e-01 3.439e-02 3.943 8.72e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03766 on 823 degrees of freedom
## Multiple R-squared: 0.02398, Adjusted R-squared: 0.02042
## F-statistic: 6.74 on 3 and 823 DF, p-value: 0.0001708
##
##
## Value of test-statistic is: -2.2711 1.9031 2.8391
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Below are the results for the U.S. / U.K. Foreign Exchange Rate time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.090954 -0.013307 0.000334 0.014569 0.076436
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.270e-02 5.089e-03 2.496 0.01285 *
## z.lag.1 -1.938e-02 6.728e-03 -2.881 0.00412 **
## tt -1.051e-05 7.056e-06 -1.489 0.13700
## z.diff.lag 3.496e-01 4.003e-02 8.735 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0223 on 548 degrees of freedom
## Multiple R-squared: 0.1287, Adjusted R-squared: 0.1239
## F-statistic: 26.98 on 3 and 548 DF, p-value: 2.749e-16
##
##
## Value of test-statistic is: -2.8808 2.9933 4.1515
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Appendix B - KPSS Test Results
Below are the results for the Real Personal Consumption Expenditures per Capita time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.5061
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the S&P 500 Index time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 14 lags.
##
## Value of test-statistic is: 8.7998
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the Gross Domestic Product: Implicit Price Deflator time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.6563
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the 10-year Treasury Constant Maturity Rate time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 13 lags.
##
## Value of test-statistic is: 20.1984
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the Unemployment Rate time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 6 lags.
##
## Value of test-statistic is: 0.5517
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the U.S. / U.K. Foreign Exchange Rate time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 6 lags.
##
## Value of test-statistic is: 0.6976
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Appendix C - ERS Test Results
Below are the results for the Real Personal Consumption Expenditures per Capita time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.044947 -0.003511 0.000151 0.004147 0.035745
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.01773 0.01037 -1.710 0.0885 .
## yd.diff.lag1 0.06147 0.05990 1.026 0.3057
## yd.diff.lag2 0.37416 0.05978 6.259 1.52e-09 ***
## yd.diff.lag3 0.03572 0.06041 0.591 0.5549
## yd.diff.lag4 -0.13413 0.06021 -2.228 0.0267 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.007617 on 270 degrees of freedom
## Multiple R-squared: 0.1406, Adjusted R-squared: 0.1247
## F-statistic: 8.834 on 5 and 270 DF, p-value: 8.828e-08
##
##
## Value of test-statistic is: -1.7097
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the S&P 500 Index time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.228556 -0.004328 0.000255 0.004714 0.105622
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.0004539 0.0002347 -1.934 0.053165 .
## yd.diff.lag1 0.0288177 0.0076961 3.744 0.000181 ***
## yd.diff.lag2 -0.0423607 0.0076993 -5.502 3.81e-08 ***
## yd.diff.lag3 0.0031176 0.0076992 0.405 0.685534
## yd.diff.lag4 -0.0079978 0.0076958 -1.039 0.298708
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.009683 on 16882 degrees of freedom
## Multiple R-squared: 0.002818, Adjusted R-squared: 0.002523
## F-statistic: 9.542 on 5 and 16882 DF, p-value: 4.19e-09
##
##
## Value of test-statistic is: -1.9337
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the Gross Domestic Product: Implicit Price Deflator time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0213168 -0.0017624 -0.0002899 0.0013188 0.0192639
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.001913 0.001655 -1.156 0.249
## yd.diff.lag1 0.585690 0.059660 9.817 <2e-16 ***
## yd.diff.lag2 0.153470 0.067096 2.287 0.023 *
## yd.diff.lag3 0.153741 0.067217 2.287 0.023 *
## yd.diff.lag4 -0.041079 0.059216 -0.694 0.488
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.003759 on 270 degrees of freedom
## Multiple R-squared: 0.6416, Adjusted R-squared: 0.635
## F-statistic: 96.68 on 5 and 270 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -1.156
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the 10-year Treasury Constant Maturity Rate time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.185364 -0.004991 0.000044 0.004832 0.095975
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.0001999 0.0001886 -1.060 0.289070
## yd.diff.lag1 0.0285763 0.0085258 3.352 0.000805 ***
## yd.diff.lag2 -0.0203729 0.0085296 -2.388 0.016931 *
## yd.diff.lag3 0.0061961 0.0085298 0.726 0.467602
## yd.diff.lag4 -0.0185160 0.0085266 -2.172 0.029906 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01269 on 13757 degrees of freedom
## Multiple R-squared: 0.001624, Adjusted R-squared: 0.001261
## F-statistic: 4.475 on 5 and 13757 DF, p-value: 0.0004478
##
##
## Value of test-statistic is: -1.0602
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the Unemployment Rate time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.210989 -0.017983 0.000577 0.020270 0.214688
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.013418 0.004138 -3.243 0.00123 **
## yd.diff.lag1 0.046275 0.034357 1.347 0.17838
## yd.diff.lag2 0.214469 0.033982 6.311 4.53e-10 ***
## yd.diff.lag3 0.175445 0.034112 5.143 3.38e-07 ***
## yd.diff.lag4 0.081097 0.034363 2.360 0.01851 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03569 on 819 degrees of freedom
## Multiple R-squared: 0.1169, Adjusted R-squared: 0.1115
## F-statistic: 21.68 on 5 and 819 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -3.2428
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the U.S. / U.K. Foreign Exchange Rate time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.091866 -0.013051 0.000154 0.013743 0.078042
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.014189 0.005862 -2.421 0.01582 *
## yd.diff.lag1 0.390715 0.042671 9.157 < 2e-16 ***
## yd.diff.lag2 -0.134890 0.045600 -2.958 0.00323 **
## yd.diff.lag3 0.127930 0.045522 2.810 0.00513 **
## yd.diff.lag4 -0.017795 0.042907 -0.415 0.67850
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02218 on 544 degrees of freedom
## Multiple R-squared: 0.1447, Adjusted R-squared: 0.1369
## F-statistic: 18.41 on 5 and 544 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -2.4206
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Appendix D - ADF Test Results over First Differences
Below are the results for the Real Personal Consumption Expenditures per Capita time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.045698 -0.003272 -0.000092 0.003996 0.037238
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.776e-03 1.045e-03 3.612 0.000361 ***
## z.lag.1 -6.228e-01 7.742e-02 -8.045 2.66e-14 ***
## tt -3.776e-06 5.825e-06 -0.648 0.517332
## z.diff.lag -3.149e-01 5.728e-02 -5.498 8.83e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.007729 on 273 degrees of freedom
## Multiple R-squared: 0.51, Adjusted R-squared: 0.5047
## F-statistic: 94.73 on 3 and 273 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -8.0446 21.5726 32.3589
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.98 -3.42 -3.13
## phi2 6.15 4.71 4.05
## phi3 8.34 6.30 5.36Below are the results for the S&P 500 Index time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.228779 -0.004370 0.000197 0.004679 0.106307
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.404e-04 1.491e-04 2.283 0.0224 *
## z.lag.1 -1.014e+00 1.072e-02 -94.527 < 2e-16 ***
## tt -5.245e-09 1.528e-08 -0.343 0.7315
## z.diff.lag 4.215e-02 7.689e-03 5.482 4.27e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.009684 on 16885 degrees of freedom
## Multiple R-squared: 0.4873, Adjusted R-squared: 0.4872
## F-statistic: 5349 on 3 and 16885 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -94.5266 2978.427 4467.64
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Below are the results for the Gross Domestic Product: Implicit Price Deflator time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0200435 -0.0017216 -0.0001678 0.0014666 0.0182528
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.631e-03 6.093e-04 2.677 0.007885 **
## z.lag.1 -1.834e-01 4.027e-02 -4.555 7.91e-06 ***
## tt -1.738e-06 2.967e-06 -0.586 0.558477
## z.diff.lag -2.271e-01 5.882e-02 -3.860 0.000141 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.003892 on 273 degrees of freedom
## Multiple R-squared: 0.165, Adjusted R-squared: 0.1559
## F-statistic: 17.99 on 3 and 273 DF, p-value: 1.11e-10
##
##
## Value of test-statistic is: -4.5548 6.9432 10.3892
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.98 -3.42 -3.13
## phi2 6.15 4.71 4.05
## phi3 8.34 6.30 5.36Below are the results for the 10-year Treasury Constant Maturity Rate time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.184675 -0.005019 -0.000115 0.004750 0.096254
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.212e-04 2.165e-04 1.022 0.3070
## z.lag.1 -9.921e-01 1.189e-02 -83.458 <2e-16 ***
## tt -3.724e-08 2.724e-08 -1.367 0.1716
## z.diff.lag 2.014e-02 8.525e-03 2.362 0.0182 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0127 on 13760 degrees of freedom
## Multiple R-squared: 0.4864, Adjusted R-squared: 0.4863
## F-statistic: 4344 on 3 and 13760 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -83.458 2321.746 3482.618
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Below are the results for the Unemployment Rate time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.222102 -0.019512 -0.000498 0.020662 0.223303
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.112e-03 2.567e-03 0.433 0.665
## z.lag.1 -6.732e-01 4.464e-02 -15.082 < 2e-16 ***
## tt -2.414e-06 5.369e-06 -0.450 0.653
## z.diff.lag -2.303e-01 3.376e-02 -6.821 1.76e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03675 on 822 degrees of freedom
## Multiple R-squared: 0.4673, Adjusted R-squared: 0.4653
## F-statistic: 240.3 on 3 and 822 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -15.0816 75.8393 113.756
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Below are the results for the U.S. / U.K. Foreign Exchange Rate time series:
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.089502 -0.013399 0.000433 0.014155 0.082267
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.650e-04 1.916e-03 -0.504 0.6147
## z.lag.1 -7.247e-01 4.890e-02 -14.820 <2e-16 ***
## tt 3.896e-07 5.995e-06 0.065 0.9482
## z.diff.lag 9.794e-02 4.257e-02 2.301 0.0218 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02238 on 547 degrees of freedom
## Multiple R-squared: 0.3363, Adjusted R-squared: 0.3327
## F-statistic: 92.39 on 3 and 547 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -14.8195 73.2069 109.8102
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.96 -3.41 -3.12
## phi2 6.09 4.68 4.03
## phi3 8.27 6.25 5.34Appendix E - KPSS Test Results over First Differences
Below are the results for the Real Personal Consumption Expenditures per Capita time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.0867
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the S&P 500 Index time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 14 lags.
##
## Value of test-statistic is: 0.0655
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the Gross Domestic Product: Implicit Price Deflator time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.5106
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the 10-year Treasury Constant Maturity Rate time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 13 lags.
##
## Value of test-statistic is: 0.0254
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the Unemployment Rate time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 6 lags.
##
## Value of test-statistic is: 0.0278
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Below are the results for the U.S. / U.K. Foreign Exchange Rate time series:
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 6 lags.
##
## Value of test-statistic is: 0.055
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216Appendix F - ERS Test Results over First Differences
Below are the results for the Real Personal Consumption Expenditures per Capita time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.049531 -0.004098 -0.000192 0.003198 0.032589
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.55916 0.09178 -6.093 3.83e-09 ***
## yd.diff.lag1 -0.34986 0.08845 -3.955 9.78e-05 ***
## yd.diff.lag2 0.05094 0.08636 0.590 0.556
## yd.diff.lag3 0.09321 0.08103 1.150 0.251
## yd.diff.lag4 -0.02976 0.06033 -0.493 0.622
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.007794 on 269 degrees of freedom
## Multiple R-squared: 0.5079, Adjusted R-squared: 0.4988
## F-statistic: 55.53 on 5 and 269 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -6.0926
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the S&P 500 Index time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.216997 -0.006451 -0.001622 0.003481 0.131864
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.390294 0.011289 -34.57 <2e-16 ***
## yd.diff.lag1 -0.451084 0.011585 -38.94 <2e-16 ***
## yd.diff.lag2 -0.368314 0.011097 -33.19 <2e-16 ***
## yd.diff.lag3 -0.234705 0.009854 -23.82 <2e-16 ***
## yd.diff.lag4 -0.117211 0.007643 -15.34 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0103 on 16881 degrees of freedom
## Multiple R-squared: 0.4198, Adjusted R-squared: 0.4196
## F-statistic: 2443 on 5 and 16881 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -34.5717
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the Gross Domestic Product: Implicit Price Deflator time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0225613 -0.0017828 -0.0002331 0.0013980 0.0182848
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.14752 0.04007 -3.682 0.00028 ***
## yd.diff.lag1 -0.27011 0.06529 -4.137 4.71e-05 ***
## yd.diff.lag2 -0.09393 0.06583 -1.427 0.15473
## yd.diff.lag3 0.06496 0.06427 1.011 0.31306
## yd.diff.lag4 0.04599 0.05916 0.778 0.43754
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.003774 on 269 degrees of freedom
## Multiple R-squared: 0.1701, Adjusted R-squared: 0.1546
## F-statistic: 11.03 on 5 and 269 DF, p-value: 1.136e-09
##
##
## Value of test-statistic is: -3.6818
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the 10-year Treasury Constant Maturity Rate time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.184942 -0.002324 0.003321 0.007816 0.094627
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.598998 0.015010 -39.908 < 2e-16 ***
## yd.diff.lag1 -0.283956 0.014337 -19.806 < 2e-16 ***
## yd.diff.lag2 -0.219314 0.013160 -16.666 < 2e-16 ***
## yd.diff.lag3 -0.126455 0.011313 -11.177 < 2e-16 ***
## yd.diff.lag4 -0.059233 0.008513 -6.958 3.6e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01326 on 13756 degrees of freedom
## Multiple R-squared: 0.4398, Adjusted R-squared: 0.4396
## F-statistic: 2160 on 5 and 13756 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -39.9077
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the Unemployment Rate time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.24724 -0.02310 -0.00227 0.01716 0.18477
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.07890 0.02279 -3.462 0.000564 ***
## yd.diff.lag1 -0.80731 0.03840 -21.021 < 2e-16 ***
## yd.diff.lag2 -0.53577 0.04550 -11.776 < 2e-16 ***
## yd.diff.lag3 -0.31915 0.04411 -7.236 1.07e-12 ***
## yd.diff.lag4 -0.18193 0.03383 -5.378 9.84e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03689 on 818 degrees of freedom
## Multiple R-squared: 0.4577, Adjusted R-squared: 0.4544
## F-statistic: 138.1 on 5 and 818 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -3.462
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Below are the results for the U.S. / U.K. Foreign Exchange Rate time series:
##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.092914 -0.015074 -0.001507 0.012543 0.084508
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.65376 0.06836 -9.564 <2e-16 ***
## yd.diff.lag1 0.04966 0.06478 0.767 0.444
## yd.diff.lag2 -0.08292 0.05920 -1.401 0.162
## yd.diff.lag3 0.04119 0.05019 0.821 0.412
## yd.diff.lag4 0.03543 0.04325 0.819 0.413
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02241 on 543 degrees of freedom
## Multiple R-squared: 0.3398, Adjusted R-squared: 0.3337
## F-statistic: 55.89 on 5 and 543 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -9.564
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57Appendix G - General Code for any D Differences - ADF, KPSS, and ERS tests
D <- 2 # Number of Differences to take
# ADF Test - D Differences
for(i in 1:length(names.of.data)){
# ADF on D Difference of Data
if(log.transform.dummy[i] == 0){ ADF.data.name <- paste("ADF.diff.D.", letters[i], sep = "")
ADF.values <- summary(
ur.df(diff(eval(parse(text = paste("data.", letters[i], sep = ""))), differences = D),
type = "trend", selectlags = "BIC"))
assign(ADF.data.name, ADF.values)
rm(ADF.data.name)
rm(ADF.values)}
# ADF on Log-Transformed D Difference of Data
if(log.transform.dummy[i] == 1){ ADF.data.name <- paste("ADF.diff.D.", letters[i], sep = "")
ADF.values <- summary(
ur.df(diff(eval(parse(text = paste("log.data.", letters[i], sep = ""))), differences = D),
type = "trend", selectlags = "BIC"))
assign(ADF.data.name, ADF.values)
rm(ADF.data.name)
rm(ADF.values)}
}
# KPSS Test - D Differences
for(i in 1:length(names.of.data)){
# KPSS on D Difference of Original Data
if(log.transform.dummy[i] == 0){ KPSS.data.name <- paste("KPSS.diff.D.", letters[i], sep = "")
KPSS.values <- summary(
ur.kpss(diff(eval(parse(text = paste("data.", letters[i], sep = ""))), differences = D),
type = "tau", lags = "short"))
assign(KPSS.data.name, KPSS.values)
rm(KPSS.data.name)
rm(KPSS.values)}
# KPSS on D Difference of Log-Transformed Data
if(log.transform.dummy[i] == 1){ KPSS.data.name <- paste("KPSS.diff.D.", letters[i], sep = "")
KPSS.values <- summary(
ur.kpss(diff(eval(parse(text = paste("log.data.", letters[i], sep = ""))), differences = D),
type = "tau", lags = "short"))
assign(KPSS.data.name, KPSS.values)
rm(KPSS.data.name)
rm(KPSS.values)}
}
# ERS Test - D Differences
for(i in 1:length(names.of.data)){
# ERS on D Difference of Original Data
if(log.transform.dummy[i] == 0){ ERS.data.name <- paste("ERS.diff.D.", letters[i], sep = "")
ERS.values <- summary(
ur.ers(diff(eval(parse(text = paste("data.", letters[i], sep = ""))), differences = D),
type = "DF-GLS", model = "trend"))
assign(ERS.data.name, ERS.values)
rm(ERS.data.name)
rm(ERS.values)}
# ERS on D Difference of Log-Transformed Data
if(log.transform.dummy[i] == 1){ ERS.data.name <- paste("ERS.diff.D.", letters[i], sep = "")
ERS.values <- summary(
ur.ers(diff(eval(parse(text = paste("log.data.", letters[i], sep = ""))), differences = D),
type = "DF-GLS", model = "trend"))
assign(ERS.data.name, ERS.values)
rm(ERS.data.name)
rm(ERS.values)}
}Appendix H - Second Difference Results for GDP: Implicit Price Deflator Time Series
##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression trend
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0225963 -0.0014731 -0.0000949 0.0017024 0.0198236
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.742e-04 4.775e-04 -0.574 0.5663
## z.lag.1 -1.577e+00 9.573e-02 -16.474 <2e-16 ***
## tt 1.298e-06 2.972e-06 0.437 0.6627
## z.diff.lag 1.929e-01 5.896e-02 3.272 0.0012 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.003933 on 272 degrees of freedom
## Multiple R-squared: 0.6772, Adjusted R-squared: 0.6737
## F-statistic: 190.2 on 3 and 272 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -16.4742 90.4955 135.7342
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau3 -3.98 -3.42 -3.13
## phi2 6.15 4.71 4.05
## phi3 8.34 6.30 5.36##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 5 lags.
##
## Value of test-statistic is: 0.022
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type DF-GLS
## detrending of series with intercept and trend
##
##
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0294488 -0.0023297 -0.0004795 0.0011940 0.0162278
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## yd.lag -0.98483 0.15115 -6.515 3.57e-10 ***
## yd.diff.lag1 -0.27129 0.13758 -1.972 0.0497 *
## yd.diff.lag2 -0.27318 0.12037 -2.269 0.0240 *
## yd.diff.lag3 -0.12444 0.09442 -1.318 0.1887
## yd.diff.lag4 -0.01466 0.05876 -0.249 0.8032
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004058 on 268 degrees of freedom
## Multiple R-squared: 0.634, Adjusted R-squared: 0.6272
## F-statistic: 92.85 on 5 and 268 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -6.5154
##
## Critical values of DF-GLS are:
## 1pct 5pct 10pct
## critical values -3.48 -2.89 -2.57