/*  Data for examining the Taylor Rule.  Monthly data from January, 1984 through
    July, 2007 on Fed Funds rate, Inflation rate (annual rates) and Industrial
    Production Index.  Fed Funds and Industrial Production data was obtained from
    the Federal Reserve Board Website while the Inflation rate (using CPI, Urban Dwellers)
    was obtained from the Bureau of Labor Statistics Website.  */

data Taylor;
  input ff inf ip;
  datalines;
9.56	4.3	53.0653
9.59	4.7	53.6415
9.91	4.9	53.9027
10.29	4.6	54.199
10.32	4.3	54.3517
11.06	4.3	54.6017
11.23	4.3	54.8556
11.64	4.3	54.9615
11.3	4.3	54.843
9.99	4.3	55.0752
9.43	4.2	55.2701
8.38	4	55.476
8.35	3.5	55.2456
8.5	    3.6	55.0386
8.58	3.8	55.4836
8.27	3.6	55.3108
7.97	3.6	55.3992
7.53	3.7	55.4914
7.88	3.5	55.1706
7.9	    3.4	55.538
7.92	3.2	55.6156
7.99	3.2	55.5081
8.05	3.5	55.8942
8.27	3.8	56.1309
8.14	4	56.7912
7.86	3.2	56.4315
7.48	2.2	56.2835
6.99	1.6	56.415
6.85	1.7	56.4906
6.92	1.8	56.3058
6.56	1.7	56.6212
6.17	1.6	56.7882
5.89	1.8	56.9172
5.85	1.6	57.124
6.04	1.3	57.4026
6.91	1.2	57.9249
6.43	1.4	57.7593
6.1	    1.9	58.586
6.13	2.8	58.6208
6.37	3.7	58.9181
6.85	3.7	59.3766
6.73	3.7	59.6053
6.58	3.9	60.024
6.73	4.3	60.3065
7.22	4.3	60.6809
7.29	4.4	61.6967
6.69	4.5	62.0394
6.77	4.3	62.4401
6.83	4.1	62.3116
6.58	3.9	62.4233
6.58	3.8	62.6522
6.87	4	63.2068
7.09	4	63.139
7.51	4	63.2511
7.75	4.1	63.3399
8.01	4.1	63.4127
8.19	4.2	63.6615
8.3	    4.3	64.0278
8.35	4.2	64.2298
8.76	4.4	64.5087
9.12	4.5	65.0253
9.36	4.6	64.3472
9.85	4.9	64.2605
9.84	5	64.3407
9.81	5.3	63.7696
9.53	5.2	63.8971
9.24	5.1	63.1718
8.99	4.6	63.8273
9.02	4.4	63.6893
8.84	4.6	63.5514
8.55	4.7	63.6626
8.45	4.6	63.8002
8.23	5.2	63.6781
8.24	5.3	64.5966
8.28	5.2	64.8737
8.26	4.7	64.7165
8.18	4.4	64.8029
8.29	4.7	64.9737
8.15	4.8	64.8337
8.13	5.7	65.0299
8.2	    6.2	65.0568
8.11	6.4	64.5541
7.81	6.2	63.8352
7.31	6.3	63.3216
6.91	5.6	62.7892
6.25	5.3	62.3718
6.12	4.8	62.0154
5.91	4.8	62.2346
5.78	5	62.7107
5.9	    4.7	63.4509
5.82	4.4	63.6299
5.66	3.8	63.7836
5.45	3.4	64.4934
5.21	2.8	64.3766
4.81	3.1	64.2559
4.43	3	64.1986
4.03	2.7	63.8657
4.06	2.8	64.4515
3.98	3.2	65.0605
3.73	3.2	65.4299
3.82	3	65.8281
3.76	3	66.0675
3.25	3.2	66.6406
3.3	    3.1	66.3888
3.22	3	66.4531
3.1	    3.3	66.8713
3.09	3.1	67.1332
2.92	3	67.0312
3.02	3.3	67.7403
3.03	3.2	67.8209
3.07	3	67.6938
2.96	3.2	68.1078
3	    3.2	68.049
3.04	3	67.965
3.06	2.8	68.2157
3.03	2.8	68.2135
3.09	2.8	68.6673
2.99	2.8	69.2774
3.02	2.7	69.5733
2.96	2.8	70.0166
3.05	2.5	70.2242
3.25	2.5	70.3425
3.34	2.7	71.3088
3.56	2.4	71.8845
4.01	2.3	72.3899
4.25	2.5	72.6358
4.26	2.7	73.0191
4.47	2.9	73.5168
4.73	3	73.7681
4.76	2.6	74.5516
5.29	2.6	75.1657
5.45	2.6	76.1008
5.53	2.9	76.4579
5.92	2.9	76.4243
5.98	2.8	76.5856
6.05	3.1	76.4805
6.01	3.1	76.5016
6	    3	76.8082
5.85	2.8	76.3544
5.74	2.6	77.3097
5.8	    2.5	78.0567
5.76	2.7	77.9847
5.8	    2.6	78.1
5.6	    2.5	78.4024
5.56	2.8	77.6929
5.22	2.7	78.9049
5.31	2.8	78.6724
5.22	2.8	79.5866
5.24	2.8	80.1623
5.27	2.8	81.0545
5.4	    2.9	81.2816
5.22	2.8	81.7908
5.3	    3	82.3528
5.24	3.1	82.3411
5.31	3.3	83.0962
5.29	3.4	83.8605
5.25	3	83.9386
5.19	3	85.0966
5.39	2.8	86.1119
5.51	2.4	85.7796
5.5	    2.2	86.4615
5.56	2.2	86.9728
5.52	2.2	87.2844
5.54	2.3	88.6726
5.54	2.2	89.4698
5.5	    2.1	90.1325
5.52	1.9	91.273
5.5	    1.7	91.8026
5.56	1.6	92.598
5.51	1.4	92.596
5.49	1.4	92.3817
5.45	1.4	92.978
5.49	1.7	93.4553
5.56	1.6	92.8475
5.54	1.7	92.4124
5.55	1.6	94.9061
5.51	1.4	94.654
5.07	1.5	95.5102
4.83	1.5	95.6164
4.68	1.6	96.1742
4.63	1.7	96.592
4.76	1.7	97.4528
4.81	1.7	97.3342
4.74	2.3	97.7118
4.74	2.1	98.6636
4.76	2	98.4215
4.99	2.1	98.9558
5.07	2.3	99.5832
5.22	2.6	99.2121
5.2	    2.6	100.808
5.42	2.6	101.5626
5.3	    2.7	102.4154
5.45	2.8	102.6453
5.73	3.2	103.034
5.85	3.8	103.7685
6.02	3	104.5574
6.27	3.1	104.6515
6.53	3.7	104.8578
6.54	3.6	104.7334
6.5	    3.4	104.1551
6.52	3.5	104.6728
6.51	3.5	104.2402
6.51	3.4	103.8462
6.4	    3.4	103.1605
5.98	3.7	102.5699
5.49	3.5	102.0819
5.31	3	101.6519
4.8	    3.2	101.2924
4.21	3.6	100.5362
3.97	3.2	99.8852
3.77	2.7	99.4418
3.65	2.7	98.7524
3.07	2.6	98.4261
2.49	2.1	97.7373
2.09	1.9	97.5138
1.82	1.6	97.6983
1.73	1.2	98.1325
1.74	1.1	98.2565
1.73	1.4	99.0539
1.75	1.6	99.2831
1.75	1.2	99.8673
1.75	1.1	101.0209
1.73	1.5	100.5373
1.74	1.7	100.9357
1.75	1.5	100.9751
1.75	2	100.4555
1.34	2.3	100.9585
1.24	2.5	100.5236
1.24	2.6	101.0338
1.26	3	101.0429
1.25	3.1	101.2481
1.26	2.2	100.4811
1.26	2.1	100.4125
1.22	2.1	100.9162
1.01	2.1	101.1137
1.03	2.2	100.926
1.01	2.3	101.7671
1.01	2	101.5934
1	    1.8	102.5976
0.98	1.9	102.522
1	    2	102.5854
1.01	1.7	103.2286
1	    1.7	103.119
1	    2.3	103.714
1	    2.9	104.2974
1.03	3.2	103.6417
1.26	3	104.4774
1.43	2.7	105.028
1.61	2.5	104.8313
1.76	3.2	105.6252
1.93	3.6	105.6225
2.16	3.3	106.1945
2.28	2.9	106.6925
2.5	3	107.4049
2.63	3.1	107.1613
2.79	3.4	107.4095
3	    2.9	108.0171
3.04	2.5	108.4822
3.26	3.1	108.6262
3.5	    3.6	109.0229
3.62	4.7	108.0412
3.78	4.4	109.7383
4		3.5	110.8724
4.16	3.4	111.3651
4.29	4	112.2558
4.49	3.6	112.0493
4.59	3.4	112.5613
4.79	3.6	113.6589
4.94	4.1	113.4542
4.99	4.3	114.4758
5.24	4.2	114.9064
5.25	3.9	115.3983
5.25	2.1	115.3415
5.25	1.3	114.2717
5.25	2	114.0956
5.24	2.5	115.4099
5.25	2.1	114.6532
5.26	2.4	114.5703
5.26	2.8	115.3842
5.25	2.6	115.7881
5.25	2.7	115.7344
5.25	2.7	116.4562
5.26	2.4	117.1405
;

/*  Check out the Dallas Federal Reserve Publication "Measuring the Taylor Rule's Performance"
    by Adriana Z. Fernandez and Alex Nikolsko-Rzhevskyy that appeared in Economic Letter,
    Vol. 2, No. 6, June, 2007, pp. 1 - 8.  In particular, I am going to use the modeling
    approach, at least in part, of Clarida, Gali, and Gertler (1998).  See Table 1 in the
    Fernandez and Nikolsko-Rzhevskyy article.  */ 

/*  Here we get ready to model the output gap using a quadratic trend on the log of industrial
    production.  */

data Taylor;
  set Taylor;
  t = _n_;
  t2 = t*t;
  lip = log(ip);

  proc reg data = Taylor;
    *model lip = t;
    model lip = t t2;
	output r=ogap out=ogap;

data complete;
   set Taylor ogap;
   merge Taylor ogap;

 /*  Here we are preparing the data for different modeling approaches.  */

   data complete;
     set complete;
	   y = ff - 2 - inf;
	   x1 = inf - 2;
	   x2 = ogap;
	   ylag1 = lag(y);
	   ylag2 = lag2(y);
       dy = y - lag(y);
	   dx1 = x1 - lag(x1);
	   dx2 = x2 - lag(x2);
	   dylag1 = lag(dy);
	   dylag2 = lag2(dy);

run;

/*  This is the "traditional (old fashion)" approach for modeling time series regressions.
    In addition to the explanatory variables we put in a time trend to model the trends
    in the dependent and explanatory variables.  We also autocorrelation in the errors if 
    necessary.  Also, as suggested by Clarida, Gali, and Gertler (1998) we form a 
    "forward looking" equation by lagging the dependent variable (the Fed Funds rate_
    by one or two lags depending on the foresight of the Fed Open Market Committee members.  */

   proc reg data = complete;  
     model ylag2 = x1 x2 t/noint;
	 output r=resid out=autoc; 

	 run;

   proc arima data = autoc;
     identify var=resid;

	 run;

	 /*  The residuals look to be autocorrelated of third order because the acf of the
	     OLS residuals damps out while the pacf has 3 spikes and then cuts off.  ^*/

   proc autoreg data = complete;
       model ylag1 = x1 x2 t/noint nlag=3;
       model ylag2 = x1 x2 t/noint nlag=3;

     run;
	 
	/*  In the ylag2 equation we get the expected sign of the coefficients of the
	    Taylor Rule and they are statistically significant while we conclude that
	    the foresight of the Fed Open Market Committee is 2 months.  */

	/*  Here we try the "modern (unit root)" approach for modeling time series regressions
	    as in the Taylor Rule.  All of the variables have "unit roots" in them thus we must
	    work with the differenced form of the Taylor Rule equation.  */

	 	/*  In the first set of regressions we don't allow for autocorrelated errors while
	        in the second set of regressions we do.  It is proper to consider the second
	        set of regressions because there is a need to adjust for autocorrelated errors
	        in the regression.  Using this approach, with a 2 month foresight,  we get the
	        the correct a priori signs of the coefficients and they are statistically
	        significant.  */
 
   proc reg data = complete;
       model dy = dx1 dx2/noint;
	   model dylag1 = dx1 dx2/noint;
	   model dylag2 = dx1 dx2/noint;

	   run;

	   proc autoreg data = complete;
	   model dylag2 = dx1 dx2/noint nlag=2;
       model dylag2 = dx1 dx2/noint nlag=1;

       run;