panel data - u-shaped correlation

[David Meyer]

Hi everyone!

I am currently analysing the influence of the cost ratio of insurance companies and using several independent variable to find out, if those variable have a significant positiv or negativ impact on the cost ratio. Hereby the additional informations:

- panel data (5 years)
- using Fixed Effect
- dependent variable: cost ratio
- independent variable: company size, leverage etc....
--> xtreg costratio lnassets leverage lnage gwpgrowth risk inflation unemployment gdpgrowth, fe

However, some independent variables might have a u-shaped / inverse u-shaped correlation to the cost ratio. With xtreg fe, I only find out if the independent variable is positiv or negative correlated to the cost ratio.

How can I check, if some of the independent variables have a u-shaped / inverse u-shaped correlation? I have already seen in the forum the command utest. How can I download this test to me STATA and apply for my test?

Many thanks in advance for your support.

Bests,
David

[Clyde Schechter]

To install -utest-, run -ssc install utest-. Then read -help utest- for instructions on how to use it.

[Joaquin Carrascosa]

. utest var1 var2
last estimates not found
r(301);

I´m getting this error (both variables are byte type)

[Clyde Schechter]

From the help file for -utest-:

utest is used after estimation commands to test for the presence of a U-shaped or inverse U-shaped relationship between an explanatory variable and the outcome variable on a specific interval.

You don't show the estimation command you ran before -utest-. I'm guessing you didn't run one.

[Joaquin Carrascosa]

Ok, i didnt run any estimation command, ill try doing that tonight.
Thanks

[Joaquin Carrascosa]

. regress X Y YSQUARED,beta

      Source |       SS           df       MS      Number of obs   =     1,065
-------------+----------------------------------   F(2, 1062)      =      9.47
       Model |  67.4875327         2  33.7437664   Prob > F        =    0.0001
    Residual |  3783.54533     1,062  3.56266039   R-squared       =    0.0175
-------------+----------------------------------   Adj R-squared   =    0.0157
       Total |  3851.03286     1,064  3.61939179   Root MSE        =    1.8875

------------------------------------------------------------------------------
Yl |      Coef.   Std. Err.      t    P>|t|                     Beta
-------------+----------------------------------------------------------------
        X|   .1371571   .0412853     3.32   0.001                 .8434685
       XSQUARED|  -.0016685    .000459    -3.64   0.000                -.9229167
       _cons |   .3644402   .8816696     0.41   0.679                        .
------------------------------------------------------------------------------

. utest X XSQUARED

Specification: f(x)=x^2
Extreme point:  41.10168

Test:
     H1: Inverse U shape
 vs. H0: Monotone or U shape

-------------------------------------------------
                 |   Lower bound      Upper bound
-----------------+-------------------------------
Interval         |          25               65
Slope            |    .0537316        -.0797492
t-value          |    2.872767         -4.12661
P>|t|            |     .002075         .0000198
-------------------------------------------------

Overall test of presence of a Inverse U shape:
     t-value =      2.87
     P>|t|   =    .00207

Any idea on how to interpret these results? Does this mean there is an Inverse U shape relation between Y and X ?

[Clyde Schechter]

Any idea on how to interpret these results? Does this mean there is an Inverse U shape relation between Y and X ?

Yes, the null hypothesis of either a monotone or direct U-shape relationship has been rejected at p = 0.00207. So you are left with the alternate hypothesis of an inverse U-shape, according to this test. The point at which the relationship achieves its peak is estimated as when X = 41.10168

[Joaquin Carrascosa]

Ok, thanks for the help!

[David Meyer]

Thanks Clyde Schechter for ur help.

In the meantime, i have installed it but I dont know how to run the utest:

As Joaquin did, I have to run first the panel regression:
xtreg costratioW lnassetsW organization lnageW lineofbusiness leverageW gwpgrowthW riskW gdpgrowth inflation unemployment, fe

Because organization (stock or mutual) and line of business (life/nonlife vs. both) are omitted, i used the regression with dummy variable to get the result of all independent variables (i increased the matsize, because there are 930 companies over 5 years):

regress costratioW lnassetsW organization lnageW lineofbusiness leverageW gwpgrowthW riskW gdpgrowth inflation unemployment i.company

With those results, I know if the independent variable has a significant positiv or negativ impact on the cost ratio, but not if there is an u-shaped or inverse u-shaped relationship.

The syntax of the u test is as followed (according to help utest):

utest x f(x)

X is probably the variable costratioW in my example. But how can i get the non-linear term, either quadratic or inverse (i.e. f(x) )?

Many thanks in advance for your support!

[Clyde Schechter]

X is probably the variable costratioW in my example.

No. Follow your own example in #6. The "x" should be the predictor variable you are interested in exploring for a quadratic relationship to costratioW, and the f(x) should be another variable that is the square of that one, and which must also be included in the regression.

-utest- is, I believe, a fairly old command and as far as I know it does not support factor variable notation. So if you want to test for a quadratic relationship between lnassetsW and costratioW you would do this:

gen lnassetsWsq= lnassetsW^2
xtreg costratioW lnassetsW lnassetsWsq organization lnageW lineofbusiness leverageW gwpgrowthW riskW gdpgrowth inflation unemployment, fe
utest lnassetsW lnassetsWsq

[David Meyer]

Thank you so much for your support so far.

I used the codes, but probably the extreme point is not in the interval, so that there is no u shaped relation ship, right? How can I find out if there is an inverse u-shaped relationship?

Bests,
Rolf

[Clyde Schechter]

Well, yes, in this case the vertex of the parabola lies far from the interval, so within the interval you do not have a U-shaped relationship. There is no separate test for inverse-U shaped relationships. It's the same test; the difference is just based on the sign of the quadratic coefficient. In your case, that sign is negative, so if we did have a quadratic relationship, it would be an inverse-U. (But, again, the vertex is far from the interval, so we do not fit a quadratic model here.)

[David Meyer]

I have an additional question which came up:

I followed your instructions:

1) gen lnassetsWsq= lnassetsW^2
2) xtreg costratioW lnassetsW lnassetsWsq organization lnageW lineofbusiness leverageW gwpgrowthW riskW gdpgrowth inflation unemployment, fe
3) utest lnassetsW lnassetsWsq

When I run the FE regression with lnassetsWsq in 2), i received the following results:

(a)
My question is relating to the following variables: lnassetsW and lnassetsWsq:
(i) Should both be significant? (both are not significant)
(ii) Should lnassetsW has a negativ and lnassetsWsq a positiv value?

If both conditions ((i) and (ii)) are not true, i shouldnt even start an utest, right?

(b)
Can I skip the other control variables and just use the following three variables to test the u shape relationship?

xtreg costratioW lnassetsW lnassetsWsq

Many thanks in advance for your support.

[David Meyer]

Well, yes, in this case the vertex of the parabola lies far from the interval, so within the interval you do not have a U-shaped relationship. There is no separate test for inverse-U shaped relationships. It's the same test; the difference is just based on the sign of the quadratic coefficient. In your case, that sign is negative, so if we did have a quadratic relationship, it would be an inverse-U. (But, again, the vertex is far from the interval, so we do not fit a quadratic model here.)

(c)
Furthermore, does it make sense to increase the interval so that the extreme point is in the interval?

Many thanks for your help

[Clyde Schechter]

That depends on how you chose the interval.

If the interval already spans the data (or the important/credible part of the data), then expanding the interval means extrapolating the model, which is a really bad idea. If, however, you chose the interval highly restricted to an unrepresentative subset of the data, then expanding it to be more representative of the data wold be a good idea.