persistent serial correlation in BG test despite many lags

[Max Sweeney]

Hi there,
I have an issue with selecting the appropriate number of lags for the Augmented DF test due to the presence of serial correlation.

For context, I have monthly consumer confidence (CC) data from May2012-Jan2020. As I understand, I want to include the smallest number of lags that explains the serial correlation in my ADF test.

I am using the Breusch Godfrey test as follows:

reg  lCC
estat bgodfrey,  small lags (1/12)

However, I cannot accept H0 (no serial correlation) for any of these lags. I'm not sure that I am performing it correctly or what I can do to fix this. I've attached a screenshot of my output (apologies, I don't understand how to post output properly). I've repeated the BG test with more lags than 12 but still have serial correlation.

Thanks in advance,
Max

[Jeff Wooldridge]

Max: What you're doing here doesn't make sense for choosing lags in the ADF. You are simply testing whether the variable of interest, lCC, has serial correlation. It likely has a lot of serial correlation or you would not be testing the null that it actually follows a unit root (which means a lot of serial correlation). Notice how if you find significance at one lag then, logically, the other test should also be significant. For example, the test for two lags tests the null that the first and second order serial correlation coefficients are zero. So, once you see the result for lags(1), and it is significant, then you're done. But, remember, this isn't telling you what you want.

You should simply compute the ADF statistic using a few different choices. You only have T = 93, so I wouldn't get carried away. Without knowing more, I would try

dfuller lCC
dfuller lCC, lags(1) regress
dfuller lCC, lags(2) regress

I usually also do it "by hand" so I can jointly test the lags of the differences. For example,

reg D.lCC L.lCC L.D.lCC L2.D.lCC
test L.D.lCC L2.D.lCC

If lCC looks like it has a deterministic trend, that should be added to all commands.

JW

[Max Sweeney]

Thank you Jeff for explaining this to me. I now understand why my BG tests were consistently indicating serial correlation irrespective of lag length.

However, I am unsure of what appropriate lag length to choose for my ADF to decide if it is I(0) or I(1). I am conscious of the fact my sample size is small as you mention. I was told that I should be iteratively reducing the lag length. For example, iteratively reducing lCC from 13 lags, I find that the p values of lags 12, 9 and 4 to be <0.05. 9 and 12 reject H0, but 4 can't reject at the 10% level.

To elaborate on my project I have multiple time series (consumer confidence, petrol price, electric car sales and the number of chargepoints). I want to run a VECM on sales but this requires that all the variables are I(1). By just running the ADF tests with 2 lags all my variables are I(1). However, I am not sure that its appropriate to only use 2 lags for all when some of the larger lag coefficients are seemingly significant and result in conflicting ADF stationarity conclusions.

Many thanks,
Max