What influences inflation expectations?

This post made me wonder:

" ... according to Professor Eric Johnson, may be the frequency with which consumers are seeing higher prices. “Things that you buy more frequently and that have large percentage increases will weigh more in people’s perception of inflation,” Johnson was quoted as saying.
He elaborated in the article with the following example: a person paying an extra $25 to fill up the gas tank is reminded of that cost once a week, or more often if you count the times he or she sees a $4-per-gallon price in giant numbers on a sign. In contrast, a rent increase of $100 would only happen once a month but would have the same financial impact."

It seems like this should be easy to test:
Volatile components of the CPI should have a larger effect on inflation expectations.
The data for CPI consists of food, energy and all other items less food and energy (the assumed less volatile components).
Food and/or energy should have larger "impacts" on inflation expectations.
In a regression of inflation expectations on change in food, energy and all other prices, the coefficients of food/energy is expected to be larger than the coefficient on changes of all other prices. (I think this is correct. I realize that the size of the coefficient does not always imply that the effect is larger but in this case it should work because all the right hand side variables are measured in the same units - percent change.)
To be more precise, it is not the size of the coefficient but the average effect as measured by the coefficient multiplied by the average changes in food, energy and all other prices that gives the size of change.

At the risk of further embarassing myself since I haven't engaged in any real econometrics in years here are some results:
I use non-seasonally adjusted data on CPI from FRED and inflation expectation from Michigan's Survey of Consumers.

The following plots the mean and median expectations of percent changes in inflation with the percent change in food.

This is a plot of the same expectations with percent change in energy:


And the following is with percent change in all other prices:

Inflation expectations are of several orders of magnitude larger than actual percent changes in prices. Surprisingly changes in all other goods excluding food and energy are more variable than changes in food prices (mean percent change is 0.34 versus 0.31).

What about the results from an estimated regression equation? The mean percent change in each of the price indices multiplied by its regression coefficient is summarized below - the estimate is the middle line while the 95% confidence intervals are the top and bottom of the bars:


Surprisingly, (if my naive estimates are correct), changes in food and energy prices have smaller effects on consumer expectations of inflation than changes in all other prices.

I am also reminded by Jim Hamilton that time varying volatility might be present in the dependent variable. (In fact, it looks like it's present in all the series but I don't know how to handle this.) I reestimated the above assuming a GARCH(1,1) and find that the results do not change substantially. (I haven't plotted the coefficients yet and maybe at some point I will update this post with the plots.)