Demand Function Problem: Spending is I(1), Prices I(0)

[mschultz]

I'm estimating a demand equation of the following form, with all variables expressed as natural logarithms:

Expenditure = f(Income, Wealth, Own Price, Price of Substitute)

The model performs quite well, save for issues around the price variables. Regardless of the time period chosen (e.g., 1948-2014, 1995-2007, etc), or use alternative price variables (ex.: item CPI, ratio of item CPI and all items CPI, actual price), the price series are always I(0) in the log form, and, in some circumstances, I(2) in their non-log'd form.

Could anyone provide some guidance?

Thank you.

[startz]

If income is I(1) it's entirely plausible that expenditure is I(1) even if prices are I(0).

[mschultz]

Income and wealth are both I(1) series. My issue is with the mixing of I(1) and I(0) due to the price measures being I(0). When I was in school, it was continuously stressed that the results from a mixed order regression are not to be considered valid, hence my concerns here. I'm presently reading through a stack of papers on the topic, though:


Stewart. 2011. A note on spurious significance in regressions involving I (0) and I (1) variables.
http://link.springer.com/article/10.100 ... 010-0404-5

Stewart. 2006. Spurious correlation of I(0) regressors in models with an I(1) dependent variable.
http://www.sciencedirect.com/science/ar ... 650500385X

Pesaron et al. 2001. Bounds testing approaches to the analysis of level relationships.
http://onlinelibrary.wiley.com/doi/10.1 ... 6/abstract

Marmol. 1998. Spurious regression theory with nonstationary fractionally integrated processes.
http://www.sciencedirect.com/science/ar ... 7697000857

Marmol. 1996. Nonsense regressions between integrated processes of different orders.
http://onlinelibrary.wiley.com/doi/10.1 ... x/abstract

Marmol. 1995. Spurious regressions between I(d) processes.
http://onlinelibrary.wiley.com/doi/10.1 ... E09.f03t03

Hassler. 1994. Spurious regressions when stationary regressors are included.
http://www.sciencedirect.com/science/ar ... 6595007288

[startz]

I misunderstood the question. You're quite correct to worry.

[mschultz]

Ah. Thank you for your thoughts, in any case.

[mschultz]

The papers I am reading through are thoroughly focused upon spurious regression in the levels. My demand equation is estimated in log difference (ld) form, e.g:
ld_Expenditure = ld_Income, ld_Wealth, ld_OwnPrice, ld_SubstitutePrice

Does that change the nature of the problem?

[startz]

My memory, imperfect, is that it does. You know have stationary variables except that the price is "over-differenced." I think that's okay, but my memory is kind of hazy.

[mschultz]

Oh? Being vaguely okay would be a marked improvement from my current understanding. I'll continue reviewing, see what I can find about problems associated with over differencing.

Thank you!

[startz]

To satisfy my own curiosity I ran a very small monte carlo. It doesn't prove anything, but it suggests that you're okay.

Code: Select all

'test of I(0) on I(-1) 'Dick Startz 'July 2015 wfcreate u 1000 series price = nrnd series income = 0 series wealth = 0 smpl 2 1000 income = .1 + income (-1) + nrnd wealth = .2 + wealth (-1) + nrnd series expenditure = income + 2*wealth + 3*price + nrnd ls d(expenditure) c d(income) d(wealth) d(price)