Solved – the interpretation of (panel data) Quantile Regression

estimationpanel dataquantile regression

I estimated a (panel data) quantile regression model using qregpd in Stata 13. It is not clear what is the interpretation of the estimate. Let's say that I choose the 50th quantile, and I find that the estimate of the main parameter of interest is -0.5.

What is the interpretation of the estimate -0.5? Let's say that -0.5 is the estimated parameter for the effect of random variable x on random variable y. Let's say that x increases by 1, so that y will decrease by (-0.5 * 1) = -0.5. Is this effect true only for the guy at the median, or for all individuals?

Best Answer

You can’t really say much about the guy at the median or how individuals will respons to changes in X. What you may say is that the median of a population with X equal to x is estimated to be 0.5 lower than the median of a population with X equal to (x+1), all other things being equal.

See this answer for a thorough discussion of how to interpret the result: https://stats.stackexchange.com/a/159928

Related Solutions

Solved – Is it possible to interpret standardized beta coefficients for quantile regression

Yes, that is the interpretation. One way in which you can see this is by predicting the median for different values of your standardized, each 1 unit (in this case standard deviation) appart. Than you can look at how much these predicted medians differ, and you will see that that is exactly the same number as your standardized quantile regression coefficient. Here is an example:

. sysuse auto, clear
(1978 Automobile Data)

. 
. // standardize variables
. sum price if !missing(price,weight)

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
       price |        74    6165.257    2949.496       3291      15906

. gen double z_price = ( price - r(mean) ) / r(sd)

. 
. sum weight if !missing(price,weight)

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
      weight |        74    3019.459    777.1936       1760       4840

. gen double z_weight = ( weight - r(mean) ) / r(sd)

. 
. // estimate the quartile regression
. qreg z_price z_weight
Iteration  1:  WLS sum of weighted deviations =  47.263794

Iteration  1: sum of abs. weighted deviations =  54.018868
Iteration  2: sum of abs. weighted deviations =  43.851751

Median regression                                    Number of obs =        74
  Raw sum of deviations 48.21332 (about -.41744651)
  Min sum of deviations 43.85175                     Pseudo R2     =    0.0905

------------------------------------------------------------------------------
     z_price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    z_weight |   .2552875   .1368752     1.87   0.066    -.0175682    .5281432
       _cons |  -.3415908   .1359472    -2.51   0.014    -.6125966    -.070585
------------------------------------------------------------------------------

. 
. // predict the predicted median for z_weight
. // is -2, -1, 0, 1, 2
. drop _all

. set obs 5
obs was 0, now 5

. gen z_weight = _n - 3

. predict med
(option xb assumed; fitted values)

. list

     +----------------------+
     | z_weight         med |
     |----------------------|
  1. |       -2   -.8521658 |
  2. |       -1   -.5968783 |
  3. |        0   -.3415908 |
  4. |        1   -.0863033 |
  5. |        2    .1689841 |
     +----------------------+

. 
. // compute how much the predicted median
. // differs between cars 1 standard deviation
. // weight apart
. gen diff = med - med[_n - 1]
(1 missing value generated)

. list

     +---------------------------------+
     | z_weight         med       diff |
     |---------------------------------|
  1. |       -2   -.8521658          . |
  2. |       -1   -.5968783   .2552875 |
  3. |        0   -.3415908   .2552875 |
  4. |        1   -.0863033   .2552875 |
  5. |        2    .1689841   .2552875 |
     +---------------------------------+

Panel Data Regression – Correct Way to Deal with Multiple Fixed Effects

When you use time dummies, you don't need a time dummy for every individual separately but for every year. So this leaves you with 28 time dummies and 997 individual dummies (always omitting the first year and first individual to avoid the dummy variable trap).

The solution to your problem is much simpler than what the other answer suggested here. If you read any introductory text on panel data (you can start with these lecture notes), you should acquaint yourself with the fixed effects estimator which is sometimes referred to as the within estimator as well.

The procedure is as follows:

average each variable over time for each individual, e.g. $\overline{y}_i = \frac{1}{T}\sum_{t=1}^{29}y_{it}$ and $\overline{x}_i = \frac{1}{T}\sum_{t=1}^{29}x_{it}$
subtract this individual mean from each observation, $\tilde{y}_{it} = y_{it} - \overline{y}_{i}$ and $\tilde{x}_{it} = x_{it} - \overline{x}_i$
regress $\tilde{y}_{it}$ on $\tilde{x}_{it}$ and you year dummies, and cluster the standard errors on the individual's ID to account for serial correlation

Even though it is not very apparent, Mundlak (1978) has shown that this procedure is equivalent to including a dummy for every individual minus 1 (again omitting the first individual) as you propose it. The advantage is obvious: you don't need all those dummies when you use this three step procedure which is called the "within transformation".

Most statistical software have ready canned packages/routines for this type of estimation as it is fairly standard. In Stata you would simply declare your data to be a panel data set which allows you to use the corresponding panel data regression and data analysis commands. For example:

webuse nlswork
tsset idcode year
xtreg ln_wage age union i.year, cluster(idcode)

Where i.year automatically inserts your year dummies for the regression. So with 29 years you lose 28 degrees of freedom which isn't awful. A nice introduction to the topic is

Wooldridge, J. (2008) "Introductory Econometrics", 4th Edition, South Western College
Baltagi, B.H. (2013) "Econometric Analysis of Panel Data", 5th Edition, John Wiley & Sons
Wooldridge, J. (2010) "Econometric Analysis of Cross Section and Panel Data", 2nd Edition, MIT Press

The last reference is for advanced students.

Best Answer

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Solved – Is it possible to interpret standardized beta coefficients for quantile regression

Panel Data Regression – Correct Way to Deal with Multiple Fixed Effects

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