Dear researchers,
Recently I came across the 1st difference OLS model in a thesis. The author argued that, since based on the unit root test, the variables are stationary in 1st differences, so the author used the 1st difference of dependent and independent variables in the OLS estimation. But, the author didn't provide a proper justification of how this is applicable. I think this question is able to create a good discussion here. Kindly share your knowledge on this matter. How acceptable is this argument?
Time series data mostly is cointegrated, and we can apply cointegration only on the variables which are stationary. to check statonarity, we use unit root and based on the unit root test result, we conclude whether the data series is stationary or not. Applying OLS by using first difference simply states that the data series is stationary at first difference.
Dear Nabya Tehreem,
Thank you for the feedback. Well, how far this concept is acceptable in the literature? As I check, I could not able to find any study using this approach. Do you have any journal with you regarding this approach? if have, you can share it with me. It will be helpful for me in understanding it more.
All the cointegration techniques are actually based on this principle that they apply OLS on the first difference of the variables. But the most simple of them all is Engle Granger approach to cointegration. You can easily read it from any Econometrics book or simply find the article of Engle and Granger.
This is not how I read the cointegration literature at all following David Hendry et al. Nonstationary series can be cointegrated. Consider the consumption function C=a + bY where neither C nor Y are stationary (variables in logs). Nonetheless, the Consumption Income ratio C-Y might be stationary and the two series adjust together over time. Moreover C - Y might be a function of a third variable (say wealth). So if D is a first difference operator and you find D(C) and D(Y) are stationary and then estimate D(C) = a + b D(Y) you have thrown out the information in C - Y (a proportional control) and arguably the integral control as well. The whole point of cointegration is the series have the same order of integration and some combination of the series is stationary.
Dear Nabya Tehreem,
Thank you for your suggestion. I will read the Engle and Granger article. But, I support John Ryding's statement. It is not necessary the variables must be stationary in 1st difference to conduct the co-integration test, For example we can use ARDL method to check for co-integration even the variables are mixed results of I(0) and I(1). Correct me if i'm wrong.
Hendry et al rather more elegantly and comprehensively make the point I was trying to get across in far greater depth http://www.econ.ku.dk/okokj/papers/dfhkjfnl.pdf. Section 7 is very clear on the difference between differencing and cointegration and is clear that cointegration involves reducing a linear combination of nonstationary series to a stationary process.
Your answers are really helpful for me too Sir John. You are truely right in your statement. Apologies for my former statement.
No apologies needed. We all need to keep an open mind to new ideas or to better understanding old ones.
@Nabya Tehreem, I believe you meant integrated (nonstationary), rather than cointegrated.
The reason to difference nonstationary data is due to the possibility of a spurious regression. This is very important. In the absence of cointegration, regressing a nonstationary dependent variable on any other nonstationary predictor will result in a spurious regression.
If cointegration exists, that means there is an equilibrium relationship between the variables. Such cases it is often useful to employ a VECM (also called VEqCM). Note, Hendry now refers to vector error correction model (VECM) as a vector equilibrium correction model - it is just an updating of the terminology.
I completely agree with Dr. Ryding on Hendry's work on cointegration, it is a great point of entry on the subject. Afterward, I think looking at Johansen and Juselius work on the cointegrated VAR model is a worthwhile pursuit.
Here is a link to a short paper on estimating cointegrating parameters in the case of nonstationary data from P. Turner and myself that you might find useful:
https://www.researchgate.net/publication/233118672_Alternative_estimators_of_cointegrating_parameters_in_models_with_nonstationary_data_an_application_to_US_export_demand
Article Alternative Estimators of Cointegrating Parameters in Models...
Using 1st or 2nd difference is not important for OLS estimator. the OLS technique can be used when all variables included in the model are stationary. Therefore, before making a decision to select a technique to estimate empirical model, one must perform appropriate unit root test to understand the order of integration.
Moradi
us
All dependent variables need to be stationary unless there are cointegrating relationships. If you regress a nonstationary variable on another nonstationary variable you have a spurious regression. The right hand side variables do not have to be stationary.
Try running a long sample regression of US Population on the price of oil. You will get a significant result and high R-squared but it is a nonsense (spurious) regression.
An interesting discussion of this topic can be found here:
http://johnhcochrane.blogspot.it/2012/01/brief-parable-of-over-differencing.html
http://faculty.chicagobooth.edu/john.cochrane/research/papers/overdifferencing.pdf
Stationarity is not enough to use the OLS estimations. You need to check also the autocorrelation of the first differences. 1st difference OLS assumes that residuals of 1st difference are independent across observations, i.e. autocorrelation is zero. Stationarity only says that autocorrelation is different from +/-1, it does not say that it is zero.
if there are 2 variables and they are I(1) then u can apply enger and granger co-integration
if variable are more the 2 and there order of level is I(1) or I(2) then u have to take step towards the JJ approach
if variable are of different level then u have to move towards the ARDL
I support the preceding point of using ARDL.
I will add that when a number of variables are I(2) in a model; there is need to first check the data to be sure that all is ok. Usually, from what I have seen, many variables that are not I(0) are I(1).
Regards
Provided that the dependent variable is I [1], the ARDL procedure can be used.
Similarly, the NARDL procedure can be considered in cases where the dependent variable is I [1] and the descriptive variables are I [1]/ i [0] or i[2]. But the basic point here is that the dependent variable I [1]. If the dependent variable is i [0], OLS can be preferred.
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