How to interpret the coefficents in the "ordered logit" model with an "interaction" explanatory variable

[Mohiuddin Alamgir]

Hello all,

I have been doing the thesis where I have to do a ordered logit regression with the presence of an interaction term in the cross section data.

My variables are:

Independent Variable ( Liquidity: in the scale of 1 to 5 where 1 is lowest liquidity and 5 is Highest liquidity)
Firm_Size(Binary variable: 0 & 1)
Time (Binary variable: 0 & 1 for different time period)
Firm_Size#Time (Interaction term)
and Location (Binary variable: 0 & 1)

So, I ran this regression: and can not interpret the result. I check for the articles which explins the orderd logistic regression, But I am still confused.

Command:

ologit Liquidity Firm_Size Time Location Firm_Size#Time

Output:


Ordered logistic regression Number of obs = 1546
LR chi2(4) = 120.21
Prob > chi2 = 0.0000
Log likelihood = -2314.4224 Pseudo R2 = 0.0253

------------------------------------------------------------------------------
lqcon_rev | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Firm_Size | -.6539028 .1356143 -4.82 0.000 -.9197019 -.3881037
Time | -1.038813 .2757469 -3.77 0.000 -1.579267 -.4983587
Firm_size# Time | .7457393 .2935339 2.54 0.011 .1704233 1.321055
Location | .8475688 .097975 8.65 0.000 .6555413 1.039596
-------------+----------------------------------------------------------------
/cut1 | -2.323204 .1340933 -2.586022 -2.060386
/cut2 | -1.206339 .122628 -1.446685 -.9659925
/cut3 | .1799421 .1198248 -.0549101 .4147943
/cut4 | 1.830445 .1331184 1.569538 2.091352
------------------------------------------------------------------------------


Now my Question is How do I explain these coefficients considering the interaction term?
I would be grateful if you can help me explaining my result.

Thanks in advance,

Mohiuddin

[Clyde Schechter]

I'm going to assume you would know what to do if there were no interaction term. That is, I will not explain what the coefficients represent generically in an ordered logit.

So in the presence of interaction, the coefficient of Firm_Size is the estimated effect (in the log odds metric) of Firm_Size (contrasting Firm_Size 1 vs Firm_Size 0) conditional on Time = 0. The coefficient of Time, similarly, is the estimated effect of Time (Time 1 vs Time 0) conditional on Firm_Size = 0. The coefficient of the interaction term estimates the difference between the effect of Firm_Size (1 vs 0) when Time = 1 and the effect of Firm_Size when Time = 0. Alternatively, it is also an estimator of the difference between the effect of Time (1 vs 0) when Firm_Size = 1 and the effect of Time when Firm_size = 0. The interaction term is, for this reason, sometimes called the difference-in-differences estimate.

If you are interested in the effect of, say, Time when Firm_Size = 1, then you have to add the coefficient of Firm_Size and the coefficient of the interaction term. -lincom- can be used for that purpose.

But actually, you are probably more interested in the predicted probabilities of the 5 outcome levels under all four combinations of Firm_Size and Time, adjusted for Location, than you are in these effects on the log-odds scale. Or you may be interested in the marginal effects of Firm_Size and Time at each level of the other, again with respect to the probabilities of the 5 outcome levels. While there are ways to get these using -lincom- and -nlcom-, it is far easier to do this with -margins-.

But first you have a problem. The output you showed is clearly not the output from the command you showed, because the output mentions a predictor variable Firm_size that does not exist in the command (which refers to Firm_Size--Stata variable names are case-sensitive). The output shown also has lqcon_rev as the dependent variable, whereas the command refers to a variable called Liquidity. So first you need to get your variable names straightened out. Also, I would simplify the command to use the ## notation to automatically generate the main effects along with the interaction. So

ologit Liquidity Firm_Size##Time Location

margins Firm_Size#Time
margins, dydx(Firm_Size) at(Time = (0 1))
margins, dydx(Time) at (Firm_Size = (0 1))

and you'll be done, with output that is relatively easy to interpret. If you don't need all of those predicted probabilities or marginal effects, of course you can skip the corresponding lines of code. And if I have guessed wrong about what the actual names of your variables are, do substitute the correct ones for those.

[Clyde Schechter]

One more comment, not directly related to your question. What is that variable Location? If it is just another binary variable, then there is no problem. Or if it is come continuous coordinate (distance from a reference point, or longitude or latitude or something like that), then also no problem. But if it is a discrete variable with more than 2 levels (e.g. 1 = North America, 2 = South America, 3 = Central America, 4 = Europe...) then just throwing it into the model that way will be a serious mis-specification. Stata will think that the 1, 2, 3,... are true numeric values and will treat the variable as if it were continuous. So if it is a discrete variable with more than 2 levels, the specification in the model should be i.Location, not bare Location.

[Mohiuddin Alamgir]

Thank you very much again dear Clyde Schechter. First of all my learnig from you has two folds. Firstly, the very well interpreted solution and secondly, the logical order you use when reply to my post. However, I actually tried to use the variable name in details which actually shorter in my original analysis to give a better idea of my problem. But, I definitly made a mess in my output editing.

I wanted to find out from the above ologit, whether the small firms(Firm_Size; 1=small, 0=large) faces more liquidity constraint as the time progresses (Time, 0= 2009, 1=2013)? However, you defintly made it clear, but, I am still confuse about my research question's answer. Another thing is that, does that Location variable has any relationship with the interaction term?

Moreover, when I put the margins command it gave me the output:

"margins, dydx(Firm_Size) at(Time = (0 1))
Time ambiguous abbreviation"

I dont know exactly, why it is so difficult to understand!!

Thanks again,
Best Regards,
Mohiuddin

[Mohiuddin Alamgir]

Sorry, The Loaction variable is a dummy Variable (0,1) for Capital and outside the capital. I didnt notice your second message before replying the first one. Thanks again. Should I use the i.Location?

[Clyde Schechter]

Another thing is that, does that Location variable has any relationship with the interaction term?

I don't understand what you mean by that. Are you asking whether the difference in effects of size on the difference over time depends on whether the firm is in or outside the capital? If so, you actually need a three-way interaction analysis. Maybe -ologit Liquidity i.Firm_Size##i.Time##i.Location- and appropriate -margins- commands following that.

With Location being a 0/1 variable, you can get away with entering it as bare Location because the -ologit- results will be the same even though Stata will think it is a continuous variable and, at least so far, you haven't used Location in your -margins- commands. But it is better practice to consistently use the i. notation for any discrete variable. That way the code is clear and unambiguous, and if you later go back and revise the variable to include an extra category you won't have to then also find every reference to it in a regression model and change those too. Also, the -margins- command sometimes treats Location and i.Location differently (even if Location is a 0/1 variable)--so in order to be able to use it with -margins- correctly, you need to specify i.Location.

margins, dydx(Firm_Size) at(Time = (0 1))
Time ambiguous abbreviation

I am perplexed by this. If your data set contains multiple variables that begin with the characters Time, then that would explain Stata's inability to decide which one you are referring to in that command. But, it would have had the same problem with your -ologit- command using Time as a variable, so you wouldn't even be at that point. We have already been over the inconsistency in use of variable names in your post, so I don't feel I can really pursue this in depth based on what you have shown so far. Please show us the entire exact code and the entire exact output for both the -ologit- and the -margins- commands. Also show us the command line and output you get from -ds Time*-. In doing that, do not retype anything into the forum. Do it by copying and pasting from either the Results window or your output log into a code block so we can be certain that you haven't inadvertently made some subtle change. (See FAQ for how to create a code block.)

[Mohiuddin Alamgir]

Thank you Dear Clyde,
Sorry for the late reply. Was busy doing assignment. Anyway, I do not find how to create the code block. I am attaching my log file in the attachment. Because if I try to paste the output it becomes broken. Please find the attachment where I use lincom, and margins command.
My dependent variable is:
lqcon (an ordinal variable for liquidity constraint, starting from 0=no constraint and 4=severly constraint.
Independent variable is
MSME(1=MSME firms, 0=Large firms)
time(1=2013, 0=2007)
interaction term: MSME##time
loc_dum=(location; 0=capital city, 1= out of capital)
and the other variables which I wil describe later if you require.
I want to know that, from the given output whether the MSME(Micro and SME firms) becoming less liquidity constraint than the large firms as the time progresses (from 2007 to 2013)?

[Clyde Schechter]

Looking at the regression output, we see that the coefficient of MSME#time is negative and fairly statistically significant with p = 0.017. So this means that when MSME = 1 (small firms), the effect of time on lqcon is less than it is in large firms (where MSME = 0). Another way of looking at it, in the log odds metric, the probability of a higher value of lqcon increases from 2007 to 2013 by an amount = 1.09 (to 2 decimal places) among the large (MSME = 0) firms. But in the small (MSME = 1) firms, the increase is 1.09-0.74 (again rounding everything to 2 decimal places). So the small firms' increase in liquidity constraints is not as great as the large firms' increase in liquidity constraints.

The -margins- output was not as helpful as it could have been because you only ran it for outcome lqcon == 0. It would have been more helpful (but more voluminous) had you run it for each of the levels of lqcon levels using the -predict(outcome(j))- option, with j = 0 through 4.

But to just answer your specific question in the last line, the answer is yes, as elaborated in the first paragraph here.

[Mohiuddin Alamgir]

Thank you. My thesis is done then!!! Anyway, I have done

-predict(outcome(j))- option, with j = 0 through 4.

But the output is very complicated now. I am adding the log file for your consideration and what actually all those outcomes show to us?
Thanks in advance.

[Clyde Schechter]

Congratulations on finishing your thesis.

Let's take, as an example, the outcome at the very end of your log, the ones with lqcon==4.

Predictive margins                                Number of obs   =       1546
Model VCE    : OIM

Expression   : Pr(lqcon==4), predict(outcome (4))

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        MSME |
          0  |   .0952046   .0127371     7.47   0.000     .0702402    .1201689
          1  |   .1409065   .0092947    15.16   0.000     .1226892    .1591238
             |
        time |
          0  |   .0919264   .0096596     9.52   0.000     .0729939    .1108588
          1  |   .2144837   .0312876     6.86   0.000     .1531612    .2758062
             |
   MSME#time |
        0 0  |   .0658393   .0118468     5.56   0.000       .04262    .0890586
        0 1  |   .1624353   .0251185     6.47   0.000     .1132038    .2116667
        1 0  |    .099287   .0095818    10.36   0.000     .0805069     .118067
        1 1  |   .2288544   .0355521     6.44   0.000     .1591736    .2985351
------------------------------------------------------------------------------

The four rows at the bottom of the table above give you the predicted probabilities of a firm finding itself in the condition lqcon == 4 (severe liquidity constraint, if I recall) for each combination of MSME and Time. So, for example, for a small (MSME = 1) firm in 2013 (time = 1), there is a 22.9% probability of severe liquidity constraint. By contrast, for a small firm (MSME = 1) in 2007 (time = 0) the probability was only 9.9%.

The rows above that are actually a bit more complicated. The row with MSME = 0 (top row of the table) shows that the overall probability of a large firm's being in a severe liquidity constraint is 9.5% regardless of time and based on the actual distribution of observations for years 2007 and 2013 in the data. This figure may or may not be of any use (most likely not, I would think.) Similarly the two rows under time in the middle of the table reflect average probabilities of severe liquidity constraint in years 2007 and 2013, respectively, averaged over small and large firms based on their representation in the data. This strikes me as having a bit more relevance than the two MSME rows, but still probably not so useful.

Moving on to the next table:

Average marginal effects                          Number of obs   =       1546
Model VCE    : OIM

Expression   : Pr(lqcon==4), predict(outcome (4))
dy/dx w.r.t. : 1.MSME

1._at        : time            =           0

2._at        : time            =           1

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.MSME       |
         _at |
          1  |   .0334477   .0098288     3.40   0.001     .0141836    .0527117
          2  |   .0664191   .0244619     2.72   0.007     .0184747    .1143635
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

The first thing to note is the header that explains that 1._at means time = 0 and 2._at means time = 1. Then looking at the corresponding rows in the table itself, we see that the marginal effect of MSME on probability of severe liquidity constraint (lqcon==4) in 2007 (time = 0, 1._at) is 0.03, meaning that the probability of severe liquidity constraint in 2007 is 3.3 percentage points more for small (MSME=1) firms than for large (MSME = 0). By contrast, in 2013 (time = 1, 2._at), the probability of severe liquidity constraint is 6.6 percentage points more for small firms than for large.

The table after that similarly shows the marginal effect of time (2013 vs 2017) on probability of severe liquidity constraint among large firms (MSME = 0, 1._at) to be an increase of 9.7 percentage points, whereas over the same period the increase in probability of severe liquidity constraints in small firms was 12.96 percentage points (MSME = 1, 2._at).

So that is how you read those tables. Putting them all together, perhaps selecting the most important overall probabilities or overall effects and then making, by hand, your own table for each level of lqcon, you can directly see the probability of each level of severity of liquidity constraint in each time period in both sizes of firm, and the extents to which those probabilities differ by size and year. The fact that the results are conveyed as probabilities rather than odds or log-odds makes them easier for most people to understand (although, to be clear, and with due respect to Maarten Buis, the odds metric is just as informative when thought of multiplicatively). I have found that my presentations and reports to people without a strong quantitative background are much more readily understood when I present things this way.

Best of luck with life after thesis!

[Mohiuddin Alamgir]

Expression   : Pr(lqcon==2), predict(outcome (2))
dy/dx w.r.t. : 1.time

1._at        : MSME            =           0

2._at        : MSME            =           1

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.time       |
         _at |
          1  |   .0409415   .0158629     2.58   0.010     .0098508    .0720322
          2  |   -.003684   .0070387    -0.52   0.601    -.0174796    .0101116
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

Thank you very much for wishes and help. From the above table, lqcon is negative and insignifcant(p value of row 2, when MSME=1); therefore, can we say that, in 2013 the MSME firms faced 0.3% less liquidity constraint than 2007?

[Clyde Schechter]

From the above table, lqcon is negative and insignifcant(p value of row 2, when MSME=1); therefore, can we say that, in 2013 the MSME firms faced 0.3% less liquidity constraint than 2007?

Not quite. We can say that in 2013, the MSME firms faced a 0.3% lower probability of being at liquidity constraint level 2 than in 2007. We can also say that this difference is not assessed with sufficient precision to assert confidently that it isn't just a zero difference.

[Mohiuddin Alamgir]

Thank you so much. Thats lot of helps. I will get to you if any more clarification required. Thank you again for all your time and efforts. Best wishes.

[Mohiuddin Alamgir]

Dear Clyde Schechter,

Hope you have been doing fine.
Can I draw a graph from this above maginal analysis, having all the five effects of margins in a single graph?
That is I want to draw a graph that have the all five of lqcon(0 to 4) in a single graph, or is there any other way I can explain the marginal results in graphs?
Thank yo in advance.