# Adding Seasonal AND annual dummies?

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?

• Annual effects are inflation changes averaged over calendar year that are residual to the other fixed effects. Since the data analysis in question is considered over nearly 27 years. Doing so will help better model the cross-sectional relationship between other exposures provided there are no cross-lagged effects. Ignoring exogenous calendar effects will possibly reduce the efficiency of analyses and cause bias if the exposures are correlated with calendar time. – AdamO Jan 14 '18 at 18:52
• @AdamO wouldent you run into issues of over identification? – EconJohn Jan 14 '18 at 22:56
• With 20 years of data measured monthly, there would be 240 observations, 19 fixed effects for year and 11 fixed effects for month for a total of 30 numerator degrees of freedom. So not that I can tell. – AdamO Jan 15 '18 at 15:52
• @AdamO Makes sense. But I conceptually don't get the idea. – EconJohn Jan 15 '18 at 17:23

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 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)!

• I might disagree that splines provide an immediately better solution. As you know, splines for time series data have a tendency to perform badly toward the early and late domains of the model. – AdamO Jan 14 '18 at 18:54
• @AdamO: I agree, especially for unconstrained splines, and even constrained splines will extrapolate local linear trends near the end of the history. However, the OP's question is not about prediction (where splines are very problematic), but about isolating the effect of minimum wage increases, where boundary effects are not all that problematic, unless the minimum wage changed mostly near those boundaries. And even so, I'd argue that low-df splines will be better than a high-df discretization with 26 (!) annual dummies. – Stephan Kolassa Jan 14 '18 at 19:09
• @StephanKolassa Check out my answer on this thread – EconJohn Jan 16 '18 at 17:42

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?)

• Thanks for posting this! To be honest, I don't see how this addresses my critique above, but I'm not a reviewer for their paper ;-) – Stephan Kolassa Jan 17 '18 at 12:15
• @StephanKolassa, this is the official answer so I had to mark it as an acceptance, but it still has some questions. – EconJohn Jan 17 '18 at 17:22