/*  The following data was obtained from the website for the Stock
    and Watson textbook Introductory Econometrics (Addison-Wesley, 2007, Second 
    edition).  It is used for the chapter 1 presentation on the empirical 
    question of "Does Reducing Class Size Improve Elementary School Education?"
    This is data gathered on 420 California school districts in 1998.  This
    is, of course, a cross-section data set.  We are first going to analyze
    the data using a one-way analysis of variance (i.e. a test of difference of
    means assuming the two populations have equal variances and are normally
    distributed.)  Then later we will use multiple regression analysis, first
    using OLS and then later adjusting for heteroskedasticity by using 
    White's heteroskedasticity-consistent standard errros for the OLS coefficient
    estimates.  obs = observation number, score = district average test score (Fifth grade),
    st_ratio = average student to teacher ratio in the fifth grade classes in the district,
    expend_pupil = expenditure per 5th grade pupil in the district, english = average 
    percentage of fifth grade students learning English in the district.  */   

  Data Combined;
  input obs score st_ratio expend_pupil english;
  datalines;
1	690.8	17.89	6385	0.0
2	661.2	21.52	5099	4.6
3	643.6	18.70	5502	30.0
4	647.7	17.36	7102	0.0
5	640.8	18.67	5236	13.9
6	605.6	21.41	5580	12.4
7	606.8	19.50	5253	68.7
8	609.0	20.89	4566	47.0
9	612.5	19.95	5356	30.1
10	612.7	20.81	5036	40.3
11	615.8	21.24	4548	52.9
12	616.3	21.00	5447	54.6
13	616.3	20.60	6567	42.7
14	616.3	20.01	4819	20.5
15	616.5	18.03	5621	80.1
16	617.3	20.25	6026	49.4
17	618.1	16.98	6723	85.5
18	618.3	16.51	5590	58.9
19	619.8	22.70	5065	77.0
20	620.3	19.91	5434	49.8
21	620.5	18.33	5726	40.7
22	621.4	22.62	4542	16.2
23	621.8	19.45	5107	45.1
24	622.1	25.05	4660	39.1
25	622.6	20.68	4555	76.7
26	623.1	18.68	5415	40.5
27	623.2	22.85	4998	73.7
28	623.5	19.27	5224	70.0
29	623.6	19.25	5139	56.0
30	624.2	20.55	4614	11.1
31	624.6	20.61	5342	80.4
32	625.0	21.07	5347	63.1
33	625.3	21.54	5036	65.1
34	625.8	19.90	5117	53.4
35	626.1	21.19	5117	49.8
36	626.8	21.87	5272	35.5
37	626.9	18.33	5226	56.1
38	627.1	16.23	6517	32.4
39	627.3	19.18	4559	65.5
40	627.3	20.28	5119	53.1
41	628.3	22.99	5338	49.6
42	628.4	20.44	5090	45.1
43	628.6	19.82	5485	30.3
44	628.7	23.21	4793	52.2
45	628.8	19.27	5093	36.8
46	629.8	23.30	4360	30.3
47	630.3	21.19	5645	49.9
48	630.4	20.87	4518	13.8
49	630.5	19.02	5864	28.9
50	630.5	21.92	5258	52.8
51	631.1	20.10	5017	44.1
52	631.4	21.48	4720	35.3
53	631.8	20.07	5471	37.5
54	631.9	20.38	5615	50.4
55	632.0	22.45	5245	31.1
56	632.0	22.90	4838	18.3
57	632.2	20.50	5368	34.7
58	632.3	20.00	5526	33.3
59	632.4	22.26	4353	33.5
60	632.8	21.56	5034	38.2
61	633.0	19.48	4692	36.9
62	633.0	17.67	5607	33.0
63	633.2	21.95	4969	58.2
64	633.7	21.78	4676	17.0
65	633.9	19.14	5306	17.7
66	634.0	18.11	5694	7.3
67	634.1	20.68	5182	31.2
68	634.1	22.62	5229	16.6
69	634.1	21.79	5339	58.1
70	634.2	18.58	6056	55.9
71	634.2	21.55	4846	5.5
72	634.4	21.15	4827	14.4
73	634.5	16.63	5367	22.8
74	634.7	21.14	5743	38.8
75	634.9	19.78	4136	64.2
76	634.9	18.98	5268	25.2
77	635.0	17.67	5238	5.4
78	635.2	17.75	5463	6.1
79	635.5	15.27	6313	0.0
80	635.6	14.00	6653	0.0
81	635.6	20.60	5533	34.9
82	635.8	16.31	6119	13.1
83	636.0	21.13	5099	36.0
84	636.1	17.49	5653	34.7
85	636.5	17.89	5329	28.7
86	636.6	19.31	5930	1.8
87	636.7	20.89	4897	30.6
88	636.9	21.29	5100	59.0
89	637.0	20.20	4826	13.6
90	637.0	24.95	4079	5.0
91	637.1	18.13	5349	1.0
92	637.3	20.00	5869	0.7
93	637.7	18.73	6462	38.5
94	637.9	18.25	6232	0.0
95	638.0	18.99	4994	17.2
96	638.0	19.89	4664	9.9
97	638.2	19.38	6107	19.4
98	638.3	20.46	5324	36.5
99	638.3	22.29	5221	39.4
100	638.3	20.70	5158	28.7
101	638.5	19.06	5131	13.7
102	638.7	20.23	5279	11.3
103	639.3	19.69	4704	3.4
104	639.3	20.36	6090	15.4
105	639.3	19.75	5357	18.0
106	639.5	19.38	5145	22.7
107	639.8	22.92	4906	18.4
108	639.8	19.37	5490	16.2
109	639.8	19.16	5719	2.0
110	639.9	21.30	4436	9.6
111	640.1	18.30	4895	41.5
112	640.2	21.08	5159	9.9
113	640.5	18.79	5491	16.1
114	640.8	19.63	5172	43.5
115	640.9	19.59	4442	8.8
116	641.1	20.87	5220	39.0
117	641.4	21.12	5253	53.9
118	641.4	20.08	5519	41.1
119	641.5	19.91	5609	1.4
120	641.8	17.81	4945	35.4
121	642.2	18.13	5223	8.6
122	642.2	19.22	4757	15.3
123	642.4	18.66	5522	19.9
124	642.8	19.60	6088	3.1
125	643.0	19.28	4961	9.9
126	643.2	22.82	4880	16.1
127	643.3	18.81	6538	43.5
128	643.4	21.37	4926	45.0
129	643.4	20.02	5205	18.2
130	643.5	21.50	4907	15.5
131	643.5	15.43	5923	0.9
132	643.7	22.40	4742	7.6
133	643.7	20.13	4954	29.1
134	644.2	19.04	5500	0.1
135	644.2	17.34	5361	50.9
136	644.4	17.02	5686	11.5
137	644.5	20.80	5594	13.5
138	644.5	21.15	4693	4.7
139	644.5	18.46	5085	21.4
140	644.5	19.14	5456	12.4
141	644.7	19.41	5105	30.1
142	645.0	19.57	5520	15.9
143	645.1	21.50	5066	43.8
144	645.3	17.53	5182	0.0
145	645.5	16.43	5960	0.0
146	645.6	19.80	5125	16.7
147	645.6	17.19	5655	3.2
148	645.8	17.62	5812	0.0
149	645.8	20.13	5468	48.5
150	646.0	22.17	5213	0.8
151	646.2	19.96	4613	1.9
152	646.3	19.04	4887	13.5
153	646.4	15.22	6455	0.0
154	646.5	21.14	5125	16.7
155	646.5	19.64	5194	5.9
156	646.7	21.05	5003	36.2
157	646.9	20.18	5338	45.0
158	646.9	21.39	5161	16.7
159	647.0	20.01	5159	15.2
160	647.3	20.29	5123	34.3
161	647.3	17.67	5783	0.0
162	647.6	18.22	4843	13.1
163	647.6	20.27	4219	4.3
164	648.0	20.20	5081	39.6
165	648.2	21.38	5145	21.7
166	648.3	20.97	4674	3.3
167	648.3	20.00	4830	7.9
168	648.7	17.15	5622	39.6
169	648.9	22.35	4949	10.1
170	649.2	22.17	5101	27.5
171	649.3	18.18	5133	14.0
172	649.5	18.96	5359	8.8
173	649.7	19.75	5149	6.4
174	649.8	16.43	5373	2.4
175	650.4	16.63	6485	6.0
176	650.5	16.38	5504	8.2
177	650.6	20.07	5106	15.5
178	650.7	18.00	5635	0.0
179	650.9	19.39	4980	0.0
180	650.9	16.43	6114	0.0
181	651.2	16.73	5850	31.8
182	651.2	24.41	4548	12.8
183	651.3	18.26	5012	0.0
184	651.4	18.96	5261	3.8
185	651.5	21.04	4276	2.5
186	651.8	20.74	4566	10.7
187	651.8	18.10	6049	0.0
188	651.9	19.85	4974	1.9
189	652.0	21.60	4432	4.6
190	652.1	22.44	4925	5.3
191	652.1	23.01	4604	27.5
192	652.3	17.75	5974	15.0
193	652.3	18.29	5216	47.9
194	652.3	19.27	4882	0.0
195	652.4	22.67	4146	1.8
196	652.4	19.29	7542	0.0
197	652.5	17.36	5247	0.0
198	652.8	19.82	5170	7.3
199	653.1	20.43	5951	17.0
200	653.4	21.04	4747	10.4
201	653.5	19.92	4944	2.3
202	653.5	19.01	6306	28.2
203	653.6	23.82	4260	8.8
204	653.7	19.37	4718	7.5
205	653.8	19.83	4751	9.8
206	653.8	15.26	5653	12.5
207	653.9	17.16	5920	0.0
208	654.1	21.81	4826	31.2
209	654.2	19.07	5533	1.4
210	654.2	25.79	3926	9.6
211	654.3	18.21	5806	0.0
212	654.6	18.17	4899	24.3
213	654.8	16.97	4885	9.6
214	654.8	21.50	5140	5.9
215	654.9	20.60	5249	1.0
216	655.0	16.99	5399	18.3
217	655.1	20.78	4965	5.0
218	655.1	15.51	6210	4.3
219	655.2	19.89	4798	3.4
220	655.3	21.40	5397	32.7
221	655.3	20.50	5079	9.5
222	655.3	19.36	4734	6.3
223	655.4	17.66	4963	13.3
224	655.5	21.02	5431	3.2
225	655.7	19.06	5382	17.0
226	655.8	22.54	5724	0.9
227	655.8	21.11	4843	6.6
228	656.4	20.05	5730	0.1
229	656.5	14.20	5636	0.0
230	656.6	18.48	5672	6.9
231	656.7	18.64	7071	9.6
232	656.7	20.95	5097	0.9
233	656.8	21.09	5483	1.2
234	656.8	18.69	5643	40.1
235	657.0	20.87	5280	16.7
236	657.0	19.83	5433	22.0
237	657.2	19.75	4529	8.9
238	657.4	19.50	6040	0.9
239	657.5	18.39	6168	0.0
240	657.6	18.79	5775	3.9
241	657.7	19.77	4807	11.9
242	657.8	19.33	4841	4.0
243	657.8	21.46	4954	22.9
244	657.9	23.08	4024	2.5
245	658.0	21.06	5179	9.9
246	658.3	18.69	4385	0.5
247	658.6	20.77	4778	9.0
248	658.8	19.31	4863	6.5
249	659.1	20.13	5486	20.7
250	659.2	20.67	5486	0.2
251	659.3	22.28	4631	14.0
252	659.4	20.60	5190	8.7
253	659.4	20.83	4601	20.0
254	659.8	19.22	6736	1.4
255	659.9	17.65	5475	2.1
256	660.1	17.00	5991	28.6
257	660.1	16.50	7203	0.2
258	660.2	19.78	4778	13.7
259	660.3	22.30	4303	0.0
260	660.8	17.73	5630	0.2
261	660.9	20.45	4709	3.1
262	661.3	20.37	5630	47.4
263	661.5	20.16	4891	22.7
264	661.6	21.62	4929	29.1
265	661.6	20.56	4905	4.4
266	661.8	19.96	5157	4.8
267	661.8	21.18	4942	5.2
268	661.8	18.81	5409	1.9
269	661.9	20.58	4981	8.7
270	661.9	18.32	4877	10.5
271	662.0	18.82	5112	1.2
272	662.4	20.82	5202	0.0
273	662.4	20.00	4138	9.4
274	662.5	19.68	5744	1.4
275	662.5	19.39	5703	35.3
276	662.6	20.93	4802	9.6
277	662.6	19.94	5442	2.2
278	662.7	20.79	4522	0.4
279	662.7	19.20	4966	0.6
280	662.8	19.02	5358	0.0
281	662.9	17.62	5785	0.9
282	663.3	20.24	5073	5.5
283	663.4	19.29	5556	0.0
284	663.5	18.83	5190	0.0
285	663.8	20.34	5211	2.4
286	663.8	19.23	5284	20.8
287	663.9	17.89	5332	2.2
288	664.0	19.52	5600	13.6
289	664.0	19.08	4882	13.3
290	664.2	19.94	4700	1.9
291	664.2	18.87	5650	23.1
292	664.3	20.14	5318	0.6
293	664.4	23.56	4709	3.3
294	664.4	21.46	5143	0.0
295	664.7	19.19	5133	2.2
296	664.8	20.13	5081	18.6
297	664.9	25.80	4016	6.2
298	665.0	18.78	5374	18.5
299	665.1	19.11	5428	32.1
300	665.2	19.70	5180	4.0
301	665.3	18.62	5400	15.3
302	665.7	21.00	5726	27.7
303	665.9	20.00	5004	12.5
304	666.0	20.98	4970	12.7
305	666.0	21.64	4747	6.9
306	666.1	20.03	4543	15.0
307	666.1	19.81	5230	11.3
308	666.2	18.00	4507	9.7
309	666.2	19.36	5303	3.3
310	666.4	20.18	5546	0.7
311	666.6	21.12	4616	5.1
312	666.6	23.39	5398	5.0
313	666.7	22.18	5118	7.0
314	666.7	19.94	5008	8.0
315	666.7	17.79	4906	0.5
316	666.8	14.71	6870	2.5
317	666.8	19.04	5302	3.8
318	667.2	20.89	5182	1.3
319	667.2	19.84	4999	4.9
320	667.5	19.52	6732	0.0
321	667.5	20.69	4613	0.2
322	667.6	18.18	5479	0.0
323	668.0	18.89	5667	17.9
324	668.1	24.89	4743	4.5
325	668.4	18.58	4907	9.5
326	668.6	18.04	5700	4.0
327	668.7	17.73	5192	20.7
328	668.8	21.45	4451	7.6
329	668.9	19.92	5628	27.5
330	669.0	20.34	4923	10.9
331	669.1	22.55	4969	5.0
332	669.3	21.10	4830	14.2
333	669.3	18.20	6717	0.2
334	669.3	20.11	5206	2.1
335	669.3	19.16	4358	8.6
336	669.8	19.55	5196	0.0
337	669.8	20.89	6002	4.8
338	670.0	18.39	5091	4.0
339	670.0	19.18	5112	1.6
340	670.7	19.40	5287	3.6
341	671.3	21.68	4889	8.7
342	671.3	19.29	5118	0.0
343	671.6	20.35	5119	0.3
344	671.6	20.96	4792	1.4
345	671.7	19.46	5450	0.1
346	671.7	19.29	5211	0.0
347	671.8	20.92	4715	3.7
348	671.9	20.90	4632	0.3
349	671.9	20.60	5029	0.0
350	671.9	19.38	5695	4.5
351	672.0	19.95	4504	2.4
352	672.1	18.85	5156	0.0
353	672.3	18.12	5434	0.0
354	672.3	19.18	5255	5.4
355	672.5	22.00	4593	0.0
356	672.6	21.58	4321	6.4
357	672.7	20.39	4921	6.5
358	673.0	16.29	6906	14.3
359	673.3	18.28	5730	3.7
360	673.3	19.37	4825	0.5
361	673.5	18.91	5388	0.0
362	673.5	16.41	5134	22.7
363	673.9	15.59	5346	0.0
364	674.3	18.71	4789	0.1
365	675.4	18.33	5330	2.7
366	675.7	17.90	5371	1.7
367	676.2	18.91	5771	0.0
368	676.5	20.32	4994	0.1
369	676.6	20.02	4854	0.0
370	676.8	24.00	4393	11.0
371	676.9	17.61	5884	0.0
372	677.3	19.35	4274	0.0
373	678.0	19.68	5147	0.1
374	678.1	18.73	5352	0.0
375	678.4	15.88	7668	0.0
376	678.8	20.05	5282	15.1
377	679.4	17.99	5369	1.5
378	679.5	16.97	6429	2.6
379	679.7	19.24	5387	1.5
380	679.8	19.20	5236	4.5
381	679.8	19.60	4830	0.3
382	680.1	20.54	5089	2.5
383	680.5	18.59	5390	0.5
384	681.3	15.60	6588	11.6
385	681.3	15.29	6197	11.4
386	681.6	17.66	4645	0.0
387	681.9	17.58	6490	1.4
388	682.2	22.33	5095	2.2
389	682.5	18.75	4842	5.3
390	682.5	18.10	6315	2.8
391	682.7	20.26	5399	0.0
392	683.3	18.80	5429	17.7
393	683.4	18.77	5644	0.0
394	684.3	20.41	6060	0.0
395	684.3	18.65	5320	0.4
396	684.8	20.71	4820	6.1
397	685.0	22.00	5208	10.1
398	686.1	17.70	5860	3.4
399	686.7	21.48	4963	8.6
400	687.5	16.70	7614	12.3
401	689.1	19.58	5566	1.6
402	691.0	17.26	7040	1.5
403	691.3	17.38	6604	10.4
404	691.9	17.35	6180	6.1
405	694.0	16.26	6461	2.3
406	694.3	17.70	6415	2.5
407	694.8	20.13	5231	0.9
408	695.2	18.27	5838	2.4
409	695.3	14.54	7712	3.8
410	696.6	19.15	5593	2.0
411	698.2	17.37	5933	0.6
412	698.3	15.14	7593	2.8
413	698.4	17.84	6500	1.4
414	699.1	15.41	7217	1.2
415	700.3	18.87	5393	2.1
416	704.3	16.47	7290	6.0
417	706.8	17.86	5741	4.7
418	645.0	21.89	4403	24.3
419	672.2	20.20	4776	3.0
420	655.8	19.04	5993	5.0
;

Data Large;
  set combined;
  if st_ratio >= 20;

run;

Data Small;
  set combined;
  if st_ratio < 20;

run;

/*  Calculating the Mean and Standard Deviations of all of data. */

  title 'Summary Statistics for all of the data';
proc means data = combined;

run;

/*  Calculating the Mean and Standard Deviations for the Large Classrooms. */

  title 'Summary Statistics for the Large Classrooms where st_ratio >= 20';
proc means data = large;

run;

/*  Calculating the Mean and Standard Deviations of the Small Classrooms. */

  title 'Summary Statistics for the Small Classrooms where st_ratio < 20';
 
proc means data = small;

run;

/*  Some simple plotting of the data.  */

proc plot data = combined;
title 'Plot of score versus student-to-teacher ratio';
  plot score*st_ratio;

run;

proc plot data = combined;
title 'Plot of score versus expenditure per pupil';
  plot score*expend_pupil;

run;

proc plot data = combined;
title 'Plot of score versus percentage taking English'; 
  plot score*english;

run; 

/*  Create a Dummy Variable labeled "treat" for treatment variable of Class Size.  */

data combined;
  set combined;
    if st_ratio < 20 then treat = 1; else treat = 0;

run;

title 'Printing out the Data to make sure treatment variable is created correctly.';

proc print data = combined;

  run;

/*  Here we run "One-Way Analysis of Variance"  */

 title 'One-way Analysis of Variance Test of Class Size Hypothesis'; 
proc reg data = combined;
  model score = treat;

  run;

/*  Here we examine the effect of class size using multiple regression where we control
    for the additional variables of "expend_pupil" and "English."  We also retreive the 
    the least squares residual ("resid") and story them in the "residuals" data set. */

title 'Multiple Regression Test of Class Size Hypothesis Using OLS';

proc reg data = combined;
   model score = st_ratio expend_pupil english;
   output r = resid out = residuals;

run;

/*  Now we merge the residuals with the variables so that we can do some residual plots to see
    if there is any heteroskedasticity in the data.  If there is, this invalidates our regression
    test results.  */
 
data together;
 set combined residuals;
 merge combined residuals;

 title 'Data with residuals';

proc print data = together;
  
run;

/*  Here we produce some diagnostic residual plots.  */

proc plot data = together;
    title 'Residual plot of residuals vs. student-to-teacher ratio';
    plot resid*st_ratio;
    title 'Residual plot of residuals vs. expenditure per pupil';
    plot resid*expend_pupil;
    title 'Residual plot of residuals vs. English proficiency of students';
    plot resid*english;

run;

/*  Now we do White's Test for Heteroskedasticity. */

data together;
   set together;
   r2 = resid**2;
   st_ratio2 = st_ratio**2;
   expend_pupil2 = expend_pupil**2;
   english2 = english**2;

   /* Some More Diagnostic Residual Plots  */

proc plot data = together;
    title 'Squared Residual plot of residuals vs. student to teacher ratio';
    plot r2*st_ratio;
    title 'Squared Residual plot of residuals vs. expenditure per pupil';
    plot r2*expend_pupil;
    title 'Squared Residual plot of residuals vs. English proficiency of students';
    plot r2*english;

run;

   /*  We use the overall F-statistic of the below heteroskecasticity test equation
       to test for heteroskedasticity.  The null hypothesis is no heteroskedasticity while
       the alternative hypothesis is heteroskedastity.  */

   title 'Whites Test for Heteroskedasticity';

proc reg data = together;
    model r2 = st_ratio st_ratio2 expend_pupil expend_pupil2 english english2;
    model r2 = english english2;
    model r2 = st_ratio st_ratio2;
	model r2 = expend_pupil expend_pupil2;

	run;

	/*  Here we adjust our multiple regression results using White's Heteroskedasticity
	    Consistent Standard Errors and, in particular, use the Davidson-McKinnon Method 1
	    Modified Version of White's standard errors.  */

	title1 'Robust Regression Results with Whites Heteroskedasticity Consistent Standard Errors';
	title2 'Using the Davidson-McKinnon Modified Version 1 (HCCME = 1)';

proc model data = together;
      parms const 649.58 b1 -0.286 b2 0.00387 b3 -0.655; 
     /*  We use the previous OLS estimates as the starting values */ 
      score = const + b1*st_ratio + b2*expend_pupil + b3*english; 
      fit score / white hccme=1;

run;

/*  Finally, let's see if our functional form is correct.  We will try to some additional
    explanatory variables including squared and interaction terms of the original variables. */

data together;
   set together;
   st_expend = st_ratio*expend_pupil;
   st_english = st_ratio*english;
   expend_english = expend_pupil*english;

run;

proc model data = together;
      parms const b1 b2 b3 b4 b5 b6 b7 b8 b9; 
      
      score = const + b1*st_ratio + b2*expend_pupil + b3*english
               + b4*st_ratio2 + b5*expend_pupil2 + b6*english2
               + b7*st_expend + b8*st_english + b9*expend_english; 
      fit score / white hccme = 1;

run;

/* As it turns out two of the quadratic and one of the interaction terms are statistically 
   significant implying that the student-to-teacher ratio has first an increasing then decreasing effect
   with size, */

proc model data = together;
      parms const b1 b2 b3 b4 b5 b6 b7 b8 b9; 
      
      score = const + b1*st_ratio + b2*expend_pupil + b3*english
               + b4*st_ratio2 + b6*english2
               + b7*st_expend; 
      fit score / white hccme = 1;

run;