Empirical Example: Returns to Scale in ElectricitySupply

Walter Sosa-Escudero

Econ 507. Econometric Analysis. Spring 2009

February 10, 2009

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Classic paper: Nerlove (1963), analyzed in detail in Hayashi(2000).

Problem: returns to scale in a regulated industry (electricpower supply). See Hayashi for a discussion of the underlyinginstitutional framework.

Data: 145 in 44 states.

Variables: total costs, factor prices (wage rate, price of fuel,and the rental price of capital), and output.

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Economic problem: cannot just look at costs vs. output.Factor prices interact with output.

Cobb-Douglas technology:

Qi = A1x11i x22i x

43i

Qi: outputxj : factorsAi: firm heterogeneity: unobservable differences inproduction efficiency.

r 1 + 2 + 3 measures degree of returns to scale.

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Cost function:

TCi = r (11 22

33 )

1/rQ1/ri p1/r1i p

2/r2i p

3/r3i

Taking logs:

lnTC1 = i +1rlnQi +

1rln p1i +

2rln p2i +

3rln p3i,

i ln [r (11 22 33 )]The linear econometric model is:

lnTC1 = 1 + 2 lnQi + 3 ln p1i + 4 ln p2i + 5 ln p3i + i

with

2 = 1/r, 3 = 1/r, 4 = 2/r, 5 = 4/r. E(i), i i , so E(i) = 0.

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Role of classical assumptions

Linearity: is a consequence of the Cobb-Douglas tecnology (inlogs).

Strict exogeneity: factor prices are given to the firmindependently of efficiency. Ouput: if prices are setindependently of efficiency, Ok. If prices are set to covercosts: unobserved efficiency is related to ouput.

No multicollinearity: Ok

Firm spillovers might harm the no-serial correlationassumption.

Heteroskedasticity very likely to be present. Careful.

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

The Cobb-Douglas technology imposes the following restriction:

3 + 4 + 5 = 1,

a consequence of r 1 + 3 + 3. This is a testable linearrestriction.If we set 5 = 1 3 4, replacing in the original model, therestricted model is

ln(TC1pi3

)= 1 + 2 lnQi + 3 ln

(p1ipi3

)+ 4 ln

(p2ipi3

)+ i

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Unrestricted model:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -3.56651 1.77938 -2.004 0.047 *

lq 0.72091 0.01743 41.352 < 2e-16 ***

lpl 0.45596 0.29980 1.521 0.131

lpf 0.42581 0.10032 4.244 3.97e-05 ***

lpk -0.21515 0.33983 -0.633 0.528

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.3923 on 140 degrees of freedom

Multiple R-squared: 0.926, Adjusted R-squared: 0.9239

F-statistic: 437.9 on 4 and 140 DF, p-value: < 2.2e-16

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Restricted model:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -4.63607 0.89499 -5.180 7.52e-07 ***

lq 0.72134 0.01739 41.477 < 2e-16 ***

lpln 0.60647 0.20724 2.926 0.004 **

lpfn 0.41437 0.09878 4.195 4.81e-05 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.3915 on 141 degrees of freedom

Multiple R-squared: 0.9272, Adjusted R-squared: 0.9256

F-statistic: 598.3 on 3 and 141 DF, p-value: < 2.2e-16

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Returns to scale: 1.4 = 1/0.72091How would you test H0 : 3 + 4 + 5 = 1? (do it).R2 = 0.926 high or low?Multicolinearity

cor(cbind(lq,lpl,lpf,lpk))

lq lpl lpf lpk

lq 1.00000000 0.02350778 -0.1689466 -0.09877366

lpl 0.02350778 1.00000000 0.3337830 -0.19015184

lpf -0.16894664 0.33378301 1.0000000 0.13090722

lpk -0.09877366 -0.19015184 0.1309072 1.00000000

What do you think?

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

A FWLT approach to estimating the effect of output

Coefficients:

Estimate Std. Error t value Pr(>|t|)

lqs 0.72091 0.01719 41.94

log total cost vs. log output

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

The FWLT plot

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

FWLT Fit

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Residuals

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Test for heteroskedasticity: White test.

Estimate Std. Error t value Pr(>|t|)

(Intercept) 2.141043 14.231399 0.150 0.880638

lq -1.020659 0.374418 -2.726 0.007262 **

lpln -0.986754 6.708450 -0.147 0.883279

lpfn 0.438692 2.830667 0.155 0.877070

I(lq^2) 0.031975 0.004993 6.404 2.32e-09 ***

I(lpln^2) -0.187534 0.813741 -0.230 0.818084

I(lpfn^2) -0.221966 0.176251 -1.259 0.210068

I(lq * lpln) -0.060283 0.087892 -0.686 0.493970

I(lq * lpfn) -0.153918 0.044745 -3.440 0.000775 ***

I(lpln * lpfn) 0.061019 0.645043 0.095 0.924776

Residual standard error: 0.2805 on 135 degrees of freedom

Multiple R-squared: 0.493, Adjusted R-squared: 0.4592

F-statistic: 14.58 on 9 and 135 DF, p-value: 2.653e-16

> 145*summary(auregw)$r.squared

[1] 71.48219

> qchisq(0.95,9)

[1] 16.91898

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply

Test for heteroskedasticity: Koenker test.

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.77486 0.09999 7.749 1.56e-12 ***

lq -0.09544 0.01464 -6.517 1.14e-09 ***

Residual standard error: 0.3361 on 143 degrees of freedom

Multiple R-squared: 0.229, Adjusted R-squared: 0.2236

F-statistic: 42.48 on 1 and 143 DF, p-value: 1.142e-09

> 145*summary(auregk)$r.squared

[1] 33.20744

> qchisq(0.95,1)

[1] 3.841459

Walter Sosa-Escudero Empirical Example: Returns to Scale in Electricity Supply