Adding Seasonal AND annual dummies?

Question

I'm currently attempting to simulate a model to measure the impacts of minimum wage increases in the Canadian economy, from the most recent Bank of Canada analytical note in appendix one (page 15 in the pdf.).

We adopt a similar framework as in Aaronsson (2001) and estimate the following reduced-form equation: $$\Delta \text{CPI}_t^p=\Sigma_{i=0}^4\eta_i \Delta\text{MW}_{t-i}+\Delta \alpha_1\text{CPI}_{t-1}^{p}+\alpha_2\Delta\text{UR}_t^p+\mu^m+\mu^y+\mu^p+\mathcal{E}_t$$

where $\text{CPI}_t^p$ represents CPI inflation for province p, in month t; $\Delta\text{MW}_{t}$ is the change in the minimum in province p; and $\Delta \text{UR}_t^p$ the change in unemployment rate in province p.We also control for monthly ($\mu^m$), yearly ($\mu^y$ ) and provincial ($\mu^p$ ) fixed effects. Estimations are done with pooled provincial data over the period from January 1991 to August 2017. 11 Lagged inflation is added to capture persistence, if any, while the provincial unemployment rate controls for economic conditions.

The coefficients of interest are the $\eta_i$ because they capture the dynamics of the direct pass-through from minimum wage increases to inflation. We allow for these measures to affect CPI inflation contemporaneously and with lags but not with leads. Overall, we find little evidence of persistence in the direct pass-through as the sum of the estimated coefficients is significant only for the specification with contemporary minimum wage .

Now I understand why there are controls for provincial and seasonal (monthly) effects, but Im not sure why they had to control for annual effects.

Why is this the case?

Answer 1

I don't think they "had to" control for annual effects. I believe this is an advanced modeling technique that is technically called A Bad Idea.

What do annual dummies do? They model (I don't like the term "control for") step changes in $\Delta\text{CPI}$ from one year to the next. But not the ones that recur every year between December and January - those are already modeled via the monthly dummies. Thus, the annual dummies model "general" ups-and-downs in $\Delta\text{CPI}$.

However, and this is the problematic part, they constrain the changes (a) to be discrete, and (b) to occur exactly when the year changes. This does not sound like normal CPI behavior. Much better to use splines to model such behavior.

Of course, you could always just ask the authors of the paper. If you do, and if they did have a good reason for their model, please do post it in an answer (or even better, recommend that they post their rationale here and downvote my answer if I'm off)!

Answer 2

After emailing the people at the Bank of Canada this question, this is their response to the reason why they included each dummy variable:

Equation A1 is estimated on monthly data, so there are 11 binary variables for month fixed effects (January excluded); 9 for province effects (Newfoundland and Labrador excluded); and 26 for year fixed effects (from 1992 to 2017, 1991 excluded). These binary variables were included to try to isolate any time or province-specific effects that could have affected the elasticity between monthly change in minimum wage and monthly inflation:

• The month fixed effects control for any seasonal pattern in CPI data.

• The province fixed effects control for any province-specific effect, constant over time.

• The year fixed effects control for any year-specific effect, common to all provinces for a given year.

It seems like this dummy is added simply as an additional control for provincial effects (which is presumably not handled by provincial dummies alone?)