A simple dorecasting model. First compute partial regression coefficients using non-log data. Test for significance. Check backcast
	Then select best model and convert data to logs. Partial regression coefficients will be then elasticities.
	1	2	3	4	5	6	7	8	9	10
	Period	Sales	Temperature	Spread	(Y-Ybar)^2	(X-Xbar)^2	(Z-Zbar)^2
		Y	X	Z				(Y-Ybar)(X-Xbar)	(Y-Ybar)(Z-Zbar)	(X-Xbar)(Z-Zbar)
1	95-96	430	30	12	2,500	9	4	150	100	6
2	96-97	335	21	10	2,025	36	0	270	0	0
3	97-98	520	35	22	19,600	64	144	1120	1680	96
4	98-99	490	42	6	12,100	225	16	1650	-440	-60
5	99-00	470	37	8	8,100	100	4	900	-180	-20
6	00-01	210	20	2	28,900	49	64	1190	1360	56
7	01-02	195	8	9	34,225	361	1	3515	185	19
8	02-03	270	17	8	12,100	100	4	1100	220	20
9	03-04	400	35	6	400	64	16	160	-80	-32
#	04-05	480	25	17	10,000	4	49	-200	700	-14
	Sum	3800	270	100	129,950	1,012	302	9,855	3,545	71
	Sum/n	380	27	10	12995	101.2	30.2	985.5	354.5	7.1
	Meaning	Ybar	Xbar	Zbar	q^2y	q^2x	q^2z	qxy	qzy	qxz
Var. Class.
DEPENDENT	Ybar	Mean Y	Avg.
Independent	Xbar	MeanX	Avg.
Independent	Zbar	MeanZ	Avg.
Variance is the average squared difference between the values in a series and the mean q^2y=12,995.0; q^2x=101.2;a^2a30.2
Covariance - measures the tendency of two variables to move or vary together.1xy=985.5;qzy=354.5;qxz=7.1
Partial Regression Coefficients
b=	(qzyqxz-qxyq^2x)/((qxz)^2-q^2xq^2z
c=	qxy*qxz-qzy*q^2x/(qxz)^2-q^2q^2z
a	Ybar-bXbar-cZbar
b=	(345,5*7.1 - 985.5*30.2)/(7.1^2-101.2*30.2
c=	(985.5 * 7.1 -345.5 * 101.2 (7.1)^2 - 101.2*30.3
a=	380 - 9.0641.27 - 9.6074 * 10
Solve from matrix		Quot.	Numerator	Denominator
b =		9.064102095	-27245.15	-3005.83
c =		9.607446196	-28878.35	-3005.83
a =		39.19478147
Note q = phi
Tests
	Visual Test
	Year	Actual	Backcast
	95-96	430	426
	96-97	335	326
	97-98	520	568
	98-99	490	478
	99-00	470	451
	00-01	210	240
	01-02	195	198
	02-03	270	270
	03-04	400	414
	04-05	480	429
(see diagram)
	1	2	3	4	5	6	7	8	9
							et	et^2	(et-et-1)^2
	Period	Sales	Temperature	Spread	(Y-Ybar)^2	Yhat (estimated)	Y - Yhat	(Y-Yhat)^2
		Y	X	Z
1	95-96	430	30	12	2500	426	4	16
2	96-97	335	21	10	2025	326	9	81	25
3	97-98	520	35	22	19600	568	-48	2,304	3,249
4	98-99	490	42	6	12100	478	12	144	3,600
5	99-00	470	37	8	8100	451	19	361	49
6	00-01	210	20	2	28900	240	-30	900	2,401
7	01-02	195	8	9	34225	198	-3	9	729
8	02-03	270	17	8	12100	270	0	0	9
9	03-04	400	35	6	400	414	-14	196	196
#	04-05	480	25	17	10000	429	51	2,601	4,225
	Sum	3800	270	100	129950			6,612	14,483
	Sum/n	380	27	10	12995
Some Simple Tests
Standard Error=	SQRT (sum (Y-Yhat)^2)/n
Calc.	SQRT 6,612/10
	26
Standard error is the most widely used statistical measure of forecasting accuracy.
It forms the basis basis for confidence and interval estimates and other goodness of fit measures.
Coefficient of Determination
The multiple coefficient of determination (R2) measures the percent of variance around the mean that
is explained by the regression equation.
R2 =1 - SE^2/q^2y
R2	0.949118892
	Durbin Watson
Auto Correlation - Do forecasted relationships hold. Impact of seasonality and and nmon linearity problems.
Durbin Watson	Sum (et-et-1)^2/(e2t)
	14,483/6,612
	2.190411373
Eliminating autocorreelation by converting numbers to change.
The rule of thumb for the DW statistic between 1.5 AND 2.5 indicates no serious auto correlation
Confidence Interval Estimation
Confidence Range Y hat +/-tpSESQRT (n/(n-k)
N = number of observations
k is the number of variables including the dependent variable
tp is the value from the from the table t statistic for n - k degrees of freedom
Example:
For the sales in the example cited we want a a confidence range such that there is a probability of actual
sales falling within range. Therefore using 7 degrees of freedom this value is 1.895. Thus the confidence
interval is Yhat +/- 1.895*26*SQRT10 - 3
Yhat +/- 59
Convert to Logs so that partial regression coefficients will be elasticities
	Period	Sales	Temperature	Spread
		Y	X	Z
1	95-96	2.633468456	1.477121255	1.079181246
2	96-97	2.525044807	1.322219295	1
3	97-98	2.716003344	1.544068044	1.342422681
4	98-99	2.69019608	1.62324929	0.77815125
5	99-00	2.672097858	1.568201724	0.903089987
6	00-01	2.322219295	1.301029996	0.301029996
7	01-02	2.290034611	0.903089987	0.954242509
8	02-03	2.431363764	1.230448921	0.903089987
9	03-04	2.602059991	1.544068044	0.77815125
#	04-05	2.681241237	1.397940009	1.230448921
SUMMARY OUTPUT
Regression Statistics
Multiple R	0.985935705
R Square	0.972069215
Adjusted R Square	0.964088991
Standard Error	0.029814624
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	2	0.21655627	0.108278135	121.8097625	3.64E-06
Residual	7	0.006222383	0.000888912
Total	9	0.222778653
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1.484164282	0.069826563	21.25500971	1.28E-07	1.319050815	1.649277749	1.319050815	1.649277749
X Variable 1	0.596449831	0.046474658	12.83387227	4.05E-06	0.486554805	0.706344857	0.486554805	0.706344857
X Variable 2	0.261560184	0.035068632	7.458522634	0.000142191	0.178636106	0.344484261	0.178636106	0.344484261

Last Updated on 6/5/2005
By Brian Browne