U.S. Gasoline and Spot Market Prices

	3.0
price ($)	2.5
gas	2.0
US retail	1.5
	1.0
FOB
price
spot
WTI
Cushing, OK	10 20 30

1995 2000 2005

Fig. 9.2. Monthly US gasoline and spot market prices: January 1991–November 2006.

		9 Models with Regressor Variables
	0.5
	0.4
CCF	0.3
0.2
0.1
	0.0
	−0.1
		−20	−10
																				Lag

Fig. 9.3. Cross correlation function for gas prices and spot prices (after takinglogarithms and differencing).

the increased volatility. Also, we convert to natural logarithms to reduce the variability due to increased prices. We return to the volatility question in Chap. 19. As the spot price should be a long-term leading indicator of the retail price, this relationship could usefully be examined using cointegra-tion methods (see Tsay 2005, pp. 376–390). Given that our present focus is on shorter-term forecasting, we will eschew that approach in favor of a simpler analysis for expository purposes.

Analysis of the logarithms of spot price (Lspot) reveals that the time series may be represented as a random walk (α ˆ = 1.17 for the LL model), so that modeling in terms of first differences is a reasonable place to start; this step is consistent with the pre-whitening approach of Box et al. (1994). The cross correlation function (CCF) for the differenced series of logarithms (DLprice and DLspot) is shown in Fig. 9.3.

The CCF shows that the spot price has strong contemporaneous varia-tion with the retail price, but also that it leads retail prices by one month. Weekly data would show a more refined pattern of leading behavior given the time taken for the price of crude oil to affect pump prices. Because we are interested in forecasting, we restrict our attention to models that use the spot price with a one-month lag, Lspot(1). Models with non-zero slopes did not appear to improve matters, and we also found that a fixed seasonal pattern sufficed.

Accordingly, we consider the following models:

• Local level — ETS(A,N,N)

• Local level with seasonals — ETS(A,N,A)

• Regression on Lspot(1)

• Regression on Lspot(1) and seasonal dummies

9.3 Diagnostics for Regression Models

Table 9.2. Analysis of the US gas prices data (logarithmic transform).

Model	α	b(spot)	R ^2	s	AICR
Local level (LL)	1.71	–	0.924	0.0359	− 153.3
LL + seasonals	1.63	–	0.938	0.0339	− 156.9
Regression on Lspot(1)	–	0.531	0.756	0.0642	− 1.2
Regression on Lspot(1) + seasonals	–	0.503	0.795	0.0617	0.0
LL + Lspot(1)	1.52	0.142	0.930	0.0346	− 161.8
LL + seasonals + Lspot(1)	1.45	0.151	0.944	0.0325	− 167.6
LL + seasonals + Lspot(1) [MSOE]	1.00	0.259	0.937	0.0345	− 152.0

b(spot) denotes the slope of Lspot(1)

• Local level with regression on Lspot(1)

• Local level with seasonals and regression on Lspot(1)

• Local level with seasonals and regression on Lspot(1), and α ≤ 1

The final model is included as it corresponds to the restriction imposed on α in the multiple source of error model.

The results are summarized in Table 9.2. The AIC values are reported rel-ative to the poorest fitting model, and are labeled as AICR. Several features are apparent. First, the regression models perform poorly when the series dynamics are ignored. Second, when the local level effects are included, the coefficient of the spot price variable is much smaller. Third, the sea-sonal effects are modest but important; the values show a consistent rise in the warmer months when demand is higher. Finally, the estimates of α consistently exceed 1; to understand why this might be, we can rewrite the transition equation as

__t = y_t + (α − 1) ε _t.

From this expression, we see that α > 1 means that the price is expected to continue to move in the same direction as it did in the previous period. It is worth noting that the MSOE scheme is unable to capture such behavior.

The performance of the models could be improved by identifying outliers and making appropriate adjustments. In order to make such adjustments we now need to identify suitable diagnostic procedures.

9.3 Diagnostics for Regression Models

Hitherto, our approach to model selection has highlighted the use of infor-mation criteria, as described in Chap. 7. However, model building with regressors often involves the evaluation of many variables and multiple lags. In principle, all possible models could be evaluated by information criteria, but this approach may be computationally intensive. Also, many researchers prefer a more hands-on developmental process in which a variety of diag-nostics may be used. The diagnostics described in the rest of this section are not new, nor do they form an exhaustive list, but they cover the main

144 9 Models with Regressor Variables

questions that a researcher may wish to address. For more detailed coverage in a time series context, see Harvey (1989, Chap. 5); much of the discussion below derives from this source. For more general discussions on regression diagnostics, the reader may consult Belsley et al. (1980) particularly on multi-collinearity questions, and Cook and Weisberg (1999) on graphical methods, influence measures and outliers.

9.3.1 Goodness-of-Fit Measures

The regression residuals are defined by substituting the parameter estimates into (9.3a); we denote these sample quantities by e_t, the estimates of ε _t. After the model has been fitted, the associated degrees of freedom are n − q − 1, where q denotes the number of fitted parameters. That is, q = n_p + n_g + d, where n_p denotes the number of regression coefficients, n_g the number of parameters in the transition equations, and d the number of free states in the initial state vector x ₀.

We consider three components that provide information about the good-ness of fit. The baseline is the original (or total) sum of squared errors for the

dependent variable:

n

SST =∑(y_t − y ¯)².

t =1

We may then compute the sum of squared errors based upon fitting the innovations model alone:

n

SSE (I) =∑ e ² _t.

t =1

The coefficient of determination is then defined in the usual way as

R ^2 I =1 − SSE (I)/ SST.

(9.5)

We can then incorporate regression elements into the model and generate the sum of squared errors for the complete model [SSE(C)]. Thus, the complete model has a coefficient of determination

R ² _C =1 − SSE (C)/ SST.

The quantity R ² _C − R ² _I represents the improvement due to the inclusion of the regression terms. The efficacy of the regression component may be tested using the ratio:

_{F =} SSE (I) − SSE (C)_. SSE (C)

Under the usual null hypothesis this measure has an F distribution with (n_p, n − q − 1) degrees of freedom.

9.3 Diagnostics for Regression Models

When the local level model is used, Harvey (1989, p. 268) suggests replacing the denominator of (9.5) by the sum of squares for the first

differences

n

SSTD =∑(y_t − y_{t− 1} − y ¯_D)²,

t =1

where y ¯_D = (y_n − y ₁)/(n − 1) denotes the mean of the first differences. We could then use the modified coefficient:

R ²_D=1 − SSE (I)/ SSTD.

Intuitively, R ²_D measures the improvement of the model over a simple ran-dom walk with drift and may be used to test the value of the innovations model beyond differencing. However, because competing models may imply different orders of differencing in the reduced forms, we prefer to use the measures defined earlier.

9.3.2 Standard Errors

Approximate standard errors for individual coefficients may be determined from the information matrix; details are given in Harvey (1989, pp. 140–143).

9.3.3 Checks of the Residuals

Many diagnostics have been developed that search for structure among the residuals in the series. Again, we mention only some of the standard tests; those seeking a more detailed treatment should consult Harvey (1989, Chap. 5).

Дата добавления: 2015-10-24