IV

Paulo Fagandini

Nova SBE

Review

Review

2SLS estimator

Exercises

https://moodle.novasbe.pt/mod/folder/view.php?id=4021

Exercise 6.1

By how much taxes must be increased ot reach a certain reduction in cigarette consumption? Using the price elasticity for the demand, we are interested in estimating \(\beta_1\) in:

\[ \ln\left(Q_i^{cigarettes}\right)=\beta_0+\beta_1\ln \left(P_i^{cigarettes}\right)+u_i \]

where \(\ln \left(Q_i^{cigarettes}\right)\) is the number of cigarette packs per capita sold and \(\ln\left(P_i^{cigarettes}\right)\) is the after-tax average real price per pack of cigarettes in state \(i\). We use the data set \(CigarettesSW\) - data on cigarette consumption for the 48 continental US States in 1995.

Exercises 6.1

1. Is it reasonable to use an OLS regression of log quantity on log price to estimate the price elasticity for the demand? Why?

An OLS regression of log quantity on log price cannot be used to estimate the effect of price since there is simultaneous causality between demand and supply. A possible solution is to use \(IV\) regression.

Exercise 6.1

2. Assume \(\ln \left(SalesTax_i\right)\) is used as an instrumental variable for the endogenous regressionr, \(\ln\left(P_i^{cigarettes}\right)\). \(\ln\left(SalesTax_i\right)\) is the portion of taxes on cigarettes arising from the general sales tax. Discuss the two conditions that need to be satisfied for \(\ln\left(SalesTax_i\right)\) to be a valid instrument.

It is relevant if:

1. Relevance condition: \(X\) and its instrument \(Z\) must be correlated. The tax is included in the (after-tax) average price per pack.
2. Instrument exogeneity condition: The instrument \(Z\) must not be correlated with the error term \(u\). It is plausible that it is exogenous since the sales tax does not influence quantity sold directly but indirectly through the price.

Exercise 6.1

The output in the next slide presents estiamtes for three different specifications: 1. is a simple linear regression for the price elasticity; 2. is the first-stage of the 2SLS Estimator 3. the second stage of the 2SLS.

Exercise 6.1

Regression

Exercise 6.1

3. How much of the observed variation in \(\ln\left(P_i^{cigarettes}\right)\) is explained by the instrument \(\ln\left(SalesTax_i\right)\) Relate the result with the concept of weak instrument.

Thiso can be answered by looking at the regression’s \(R^2\) which states that about \(47%\) of the variation in after tax prices is explained by the variation of the sales tax across states. Threfore, there is evidence against being a weak instrument. A weak instrument explains little varaition int he endogenous regressor X (which might lead to inaccurate estimates as the distribution of the estiamtor deviates considerably from a normal distribution, even in large samples).

Exercise 6.1

4. Explain the difference between teh second-stage regression (3) and the SLR(1).

In (1) we have a SLR between quantity and price. While in (3) we use the estiamted fitted values obtaiend by the first stage regression and use them as the regressors for the price elasticity equation.

Exercise 6.1

5. Interpret the coefficient for (3). Compare it with (1) and comment.

The 2SLS estiamte for \(\beta_1\) suggests that an increase in cigarette prices by 1% reduces cigarette consumption by roughly 1.08%, which is fairly elastic. Lower than (1), but still not what we expect for demand of cigarettes.

Exercise 6.1

6. Intuitively, what do you point out as the main limitations of these models?

Even though the elasticity was estimated with IV regression, there still might be a bias due to omitted variables. For example, there ar economic factors, like state income, which impact the demand for cigarettes and correlate with the sales tax. Thus, a multiple IV regression approach should be considered.

Appendix

t table

df-\(\alpha/2\)	0.50	0.25	0.20	0.15	0.10	0.05	0.025	0.01	0.005	0.001	0.0005
1	0.000	1.000	1.376	1.963	3.078	6.314	12.71	31.82	63.66	318.31	636.62
2	0.000	0.816	1.061	1.386	1.886	2.920	4.303	6.965	9.925	22.327	31.599
3	0.000	0.765	0.978	1.250	1.638	2.353	3.182	4.541	5.841	10.215	12.924
4	0.000	0.741	0.941	1.190	1.533	2.132	2.776	3.747	4.604	7.173	8.610
5	0.000	0.727	0.920	1.156	1.476	2.015	2.571	3.365	4.032	5.893	6.869
6	0.000	0.718	0.906	1.134	1.440	1.943	2.447	3.143	3.707	5.208	5.959
7	0.000	0.711	0.896	1.119	1.415	1.895	2.365	2.998	3.499	4.785	5.408
8	0.000	0.706	0.889	1.108	1.397	1.860	2.306	2.896	3.355	4.501	5.041
9	0.000	0.703	0.883	1.100	1.383	1.833	2.262	2.821	3.250	4.297	4.781
10	0.000	0.700	0.879	1.093	1.372	1.812	2.228	2.764	3.169	4.144	4.587
11	0.000	0.697	0.876	1.088	1.363	1.796	2.201	2.718	3.106	4.025	4.437
12	0.000	0.695	0.873	1.083	1.356	1.782	2.179	2.681	3.055	3.930	4.318
13	0.000	0.694	0.870	1.079	1.350	1.771	2.160	2.650	3.012	3.852	4.221
14	0.000	0.692	0.868	1.076	1.345	1.761	2.145	2.624	2.977	3.787	4.140
15	0.000	0.691	0.866	1.074	1.341	1.753	2.131	2.602	2.947	3.733	4.073
16	0.000	0.690	0.865	1.071	1.337	1.746	2.120	2.583	2.921	3.686	4.015
17	0.000	0.689	0.863	1.069	1.333	1.740	2.110	2.567	2.898	3.646	3.965
18	0.000	0.688	0.862	1.067	1.330	1.734	2.101	2.552	2.878	3.610	3.922
19	0.000	0.688	0.861	1.066	1.328	1.729	2.093	2.539	2.861	3.579	3.883
20	0.000	0.687	0.860	1.064	1.325	1.725	2.086	2.528	2.845	3.552	3.850
21	0.000	0.686	0.859	1.063	1.323	1.721	2.080	2.518	2.831	3.527	3.819
22	0.000	0.686	0.858	1.061	1.321	1.717	2.074	2.508	2.819	3.505	3.792
23	0.000	0.685	0.858	1.060	1.319	1.714	2.069	2.500	2.807	3.485	3.768
24	0.000	0.685	0.857	1.059	1.318	1.711	2.064	2.492	2.797	3.467	3.745
25	0.000	0.684	0.856	1.058	1.316	1.708	2.060	2.485	2.787	3.450	3.725
26	0.000	0.684	0.856	1.058	1.315	1.706	2.056	2.479	2.779	3.435	3.707
27	0.000	0.684	0.855	1.057	1.314	1.703	2.052	2.473	2.771	3.421	3.690
28	0.000	0.683	0.855	1.056	1.313	1.701	2.048	2.467	2.763	3.408	3.674
29	0.000	0.683	0.854	1.055	1.311	1.699	2.045	2.462	2.756	3.396	3.659
30	0.000	0.683	0.854	1.055	1.310	1.697	2.042	2.457	2.750	3.385	3.646
40	0.000	0.681	0.851	1.050	1.303	1.684	2.021	2.423	2.704	3.307	3.551
60	0.000	0.679	0.848	1.045	1.296	1.671	2.000	2.390	2.660	3.232	3.460
80	0.000	0.678	0.846	1.043	1.292	1.664	1.990	2.374	2.639	3.195	3.416
100	0.000	0.677	0.845	1.042	1.290	1.660	1.984	2.364	2.626	3.174	3.390
1000	0.000	0.675	0.842	1.037	1.282	1.646	1.962	2.330	2.581	3.098	3.300
Z	0.000	0.674	0.842	1.036	1.282	1.645	1.960	2.326	2.576	3.090	3.291

F table

df2-α=0.05	1	2	3	4	5	6	7	8	9	10
1.000	161.448	199.500	215.707	224.583	230.162	233.986	236.768	238.883	240.543	241.882
2.000	18.513	19.000	19.164	19.247	19.296	19.330	19.353	19.371	19.385	19.396
3.000	10.128	9.552	9.277	9.117	9.013	8.941	8.887	8.845	8.812	8.786
4.000	7.709	6.944	6.591	6.388	6.256	6.163	6.094	6.041	5.999	5.964
5.000	6.608	5.786	5.409	5.192	5.050	4.950	4.876	4.818	4.772	4.735
6.000	5.987	5.143	4.757	4.534	4.387	4.284	4.207	4.147	4.099	4.060
7.000	5.591	4.737	4.347	4.120	3.972	3.866	3.787	3.726	3.677	3.637
8.000	5.318	4.459	4.066	3.838	3.687	3.581	3.500	3.438	3.388	3.347
9.000	5.117	4.256	3.863	3.633	3.482	3.374	3.293	3.230	3.179	3.137
10.000	4.965	4.103	3.708	3.478	3.326	3.217	3.135	3.072	3.020	2.978
11.000	4.844	3.982	3.587	3.357	3.204	3.095	3.012	2.948	2.896	2.854
12.000	4.747	3.885	3.490	3.259	3.106	2.996	2.913	2.849	2.796	2.753
13.000	4.667	3.806	3.411	3.179	3.025	2.915	2.832	2.767	2.714	2.671
14.000	4.600	3.739	3.344	3.112	2.958	2.848	2.764	2.699	2.646	2.602
15.000	4.543	3.682	3.287	3.056	2.901	2.790	2.707	2.641	2.588	2.544
16.000	4.494	3.634	3.239	3.007	2.852	2.741	2.657	2.591	2.538	2.494
17.000	4.451	3.592	3.197	2.965	2.810	2.699	2.614	2.548	2.494	2.450
18.000	4.414	3.555	3.160	2.928	2.773	2.661	2.577	2.510	2.456	2.412
19.000	4.381	3.522	3.127	2.895	2.740	2.628	2.544	2.477	2.423	2.378
20.000	4.351	3.493	3.098	2.866	2.711	2.599	2.514	2.447	2.393	2.348
21.000	4.325	3.467	3.072	2.840	2.685	2.573	2.488	2.420	2.366	2.321
22.000	4.301	3.443	3.049	2.817	2.661	2.549	2.464	2.397	2.342	2.297
23.000	4.279	3.422	3.028	2.796	2.640	2.528	2.442	2.375	2.320	2.275
24.000	4.260	3.403	3.009	2.776	2.621	2.508	2.423	2.355	2.300	2.255
25.000	4.242	3.385	2.991	2.759	2.603	2.490	2.405	2.337	2.282	2.236
26.000	4.225	3.369	2.975	2.743	2.587	2.474	2.388	2.321	2.265	2.220
27.000	4.210	3.354	2.960	2.728	2.572	2.459	2.373	2.305	2.250	2.204
28.000	4.196	3.340	2.947	2.714	2.558	2.445	2.359	2.291	2.236	2.190
29.000	4.183	3.328	2.934	2.701	2.545	2.432	2.346	2.278	2.223	2.177
30.000	4.171	3.316	2.922	2.690	2.534	2.421	2.334	2.266	2.211	2.165
40.000	4.085	3.232	2.839	2.606	2.449	2.336	2.249	2.180	2.124	2.077
60.000	4.001	3.150	2.758	2.525	2.368	2.254	2.167	2.097	2.040	1.993
120.000	3.920	3.072	2.680	2.447	2.290	2.175	2.087	2.016	1.959	1.910
1000000000.000	3.841	2.996	2.605	2.372	2.214	2.099	2.010	1.938	1.880	1.831