Testing if coefficients are statistically different from each other

[Darcy Hill]

Hi, I have run the two regressions below. The method variables are different for Dry Recycling and Compost. Is there a test that I can use to compare the coefficients these regressions for example, whether the effect of income on Dry Recycling rates is statistically different from the effect of income on Compost rates?

VARIABLES	Dry Recycling	VARIABLES	Compost
ln(income)	-1.610	ln(income)	1.092
	(2.575)		(2.023)
ln(population density)	-3.752	ln(population density)	0.186
	(2.336)		(3.571)
ln(household size)	-48.76*	ln(household size)	33.88
	(24.80)		(22.82)
Waste Method 1Wheeled Bin 100-150l	0.272	Waste Method 1Wheeled Bin 100-150l	-0.525
	(0.475)		(0.330)
1Wheeled Bin 150-250l	-0.640	1Wheeled Bin 150-250l	-0.328
	(0.401)		(0.382)
1Wheeled Bin 251-350l	-0.988*	1Wheeled Bin 251-350l	0.272
	(0.541)		(0.408)
1Wheeled Bin>350l	-0.136	1Wheeled Bin>350l	-0.00640
	(0.472)		(0.280)
1Plastic Sacks	0.569	1Plastic Sacks	0.144
	(0.590)		(0.338)
1Refuse Bins	0.259	1Refuse Bins	0.246
	(0.745)		(0.841)
1Communal Bin	-0.304	1Communal Bin	-0.0781
	(0.385)		(0.344)
1No method	-0.209	1No method	0.372
	(0.649)		(0.470)
1Other method	0.0609	1Other method	0.131
	(0.467)		(0.866)
Dry Recycling Method 2Kerbside Box<35l	-1.455***	Compost Method 3Reusable Sacks	-0.0396
	(0.539)		(0.586)
2Kerbside Box 35-50l	-0.937*	3Non Reusable Sacks	0.353
	(0.500)		(0.271)
2Kerbside Box>35l	-1.285**	3Wheeled bin<120l	0.133
	(0.510)		(0.728)
2Reusable Sacks	0.223	3Wheeled Bin 120-180l	-0.0179
	(0.547)		(0.317)
2Non Reusable Sacks	0.418	3Wheeled Bin 180-240l	1.190***
	(0.407)		(0.407)
2Wheeled Bin >120l	0.151	3Wheeled Bin 241l+	0.511
	(0.931)		(0.411)
2Wheeled Bin 120-180l	0.553	3Other	-0.0887
	(0.392)		(0.337)
2Wheeled Bin 181-240l	1.409***	Waste Average Frequency	2.221***
	(0.375)		(0.611)
2Wheeled Bin 241l+	0.412	Compost Average Frequency	-2.434***
	(0.303)		(0.653)
No Method of containment	0.182	quarter2	11.89***
	(0.697)		(0.318)
Other	0.461	quarter3	11.35***
	(0.375)		(0.316)
Waste Average Frequency	1.424**	quarter4	4.956***
	(0.570)		(0.141)
Dry Average Frequency	-0.630	Constant	-28.97*
	(0.642)		(16.34)
Quarter 2	-4.473***
	(0.122)	Observations	5,544
Quarter 3	-4.172***	Number of Local Authorities	299
	(0.118)	R-squared	0.695
Quarter 4	-2.505***
	(0.101)
Constant	91.31***
	(16.95)
Observations	5,862
Number of local authorities	311
R-squared	0.359
Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1

[Clyde Schechter]

I think you are on a fool's errand here. The notion that one can compare the effect of some variable on different outcomes is an attractive illusion, but it is an illusion nevertheless. Consider this. Suppose that a unit difference in predictor variable X is associated with a change in composting rate from 10% to 15%, and is associated with a change in recycling rate from 50 to 55%. On the one hand, the "effect" is the same in both cases: a 5 percentage point difference. On the other hand, the effect on composting is a 50% increase, but on recycling it's a 10% difference. And that's looking at it through marginal effects; you could also pose questions about the regression coefficients themselves (or derivatives of them such as odds ratios). The answers will, in general, be different.

And the above hasn't even begun to deal with other problems such as the differences in distributions of the predictor variables themselves. It's a huge mess.

At best, if you are going to go down this road, you need to be extremely clear and explicit about what you mean--which usually requires more detail than a typical audience is able to absorb and retain. Adding a "statistical test" into the mix usually just adds to the confusion and adds no information of practical value.