Testing Nonlinear Restrictions
Aaron Enriquez
Rohini Ghosh
James Inwood
The Goal
Test a hypothesis of the form:
𝐻𝑜 : 𝐶 𝛽 = 𝑞
𝐻1 : 𝐶 𝛽 ≠ 𝑞,
where C 𝛽 is a single non-linear function.
Real World Application
Long-Run Marginal Propensity to Consume
• Consumption function with different short- and long-run MPCs:
𝑙𝑛𝐶𝑡 = 𝛼 + 𝛽𝑙𝑛𝑌𝑡 + 𝛾𝑙𝑛𝐶𝑡−1 + 𝜀𝑡
• Short-run MPC is 𝛽, long-run MPC is 𝛿 =
• We could test the hypothesis 𝛿 = 1:
𝛽
1−𝛾
𝛽
𝐻𝑜 :
=1
1−𝛾
(Greene, pg. 132)
How is it done?
• Form the Wald-like statistic:
𝑊 = 𝐶 𝛽 −𝑞
′
−1
var 𝐶 𝛽
𝐶 𝛽 −𝑞
~ 𝑋 2 (𝐽)
(Aadland. Inference and Prediction. pg. 7)
That variance… gross.
var 𝐶 𝛽
Calculating the variance of a nonlinear function is
• potentially time-consuming
• potentially tricky
So, let’s approximate it!
Taylor Series Approximation
𝛿𝐶 𝛽
𝐶 𝛽 ≈𝐶 𝛽 +(
)′(𝐵 − 𝐵)
𝛿𝛽
Recall:
• The Taylor Series expansion of a function around a value 𝛼 is equal to:
𝑓′
𝑓 ′′
𝑥−𝛼 2
2!
𝑓 𝑥 =𝑓 𝛼 +
𝛼 𝑥−𝛼 +
𝛼
+⋯
• We can usually drop the higher order terms, which leaves us with:
𝑓 𝑥 ≈ 𝑓 𝛼 + 𝑓′(𝛼)(𝑥 − 𝛼)
Note:
^
^
• If plim 𝛽 =𝛽, we can use C(𝛽) as an estimate of C(𝛽) (Slutsky’s theorem)
• This is an application of the Delta Method
(Greene. (5-34). pg. 132)
Variance Estimation
𝛿𝐶 𝛽
𝐶 𝛽 = 𝐶 𝛽 +(
)′(𝛽 − 𝛽)
𝛿𝛽
𝑣𝑎𝑟[𝐶(𝛽)] ≈
𝛿𝐶 𝛽
𝛿𝛽
We can use 𝑠 2 𝑋 ′ 𝑋
′
−1
𝑣𝑎𝑟[𝛽]
𝛿𝐶 𝛽
(
𝛿𝛽
)
to estimate the variance of the estimator.
(Greene. (5-34). pg. 132)
Let’s see a math application.
Consider the nonlinear function 𝑌 = 𝑋 2 .
We know: 𝑓 𝑥 = 𝑥 2 and 𝑓 ′ 𝑥 = 2𝑥.
We then have:
𝑣𝑎𝑟 𝑌 ≈ 2𝜇𝑥 2 𝑣𝑎𝑟 𝑥 = 4𝜇𝑥2 𝜎𝑥2
Note:
• See the MATLAB Example.
(http://www.math.montana.edu/~parker/PattersonStats/Delta.pdf)
Testing a nonlinear hypothesis
• Generalize the Wald test to test a single nonlinear restriction on β:
𝐻𝑜 : 𝐶 𝛽 = 𝑞
𝐻1 : 𝐶 𝛽 ≠ 𝑞,
Where 𝐶 𝛽 is a nonlinear function of the regression coefficient(s)
• Assume
• 𝐶 𝛽 is continuously differentiable
• 𝐺=
𝜕𝐶(𝛽)
is
𝜕𝛽 ′
full rank
The Wald test
• Using our estimated variance form the Wald-statistic:
𝑊 = 𝐶 𝛽 −𝑞
′
−1
𝐸𝑠𝑡. 𝐴𝑠𝑦. Var 𝐶 𝛽
𝐶 𝛽 −𝑞
asy
~ 𝑋 2 (𝐽)
Note: if the sample size is large enough (large-sample properties)
• Test does not need assumption of normality
• Test is invariant to how the hypothesis is formulated
• Test-statistic has a chi-squared distribution with degrees of freedom equal to the number of
restrictions
Generalized Wald test (multiple restrictions)
• Let 𝐶(𝛽) be a set of 𝐽 functions of the estimated parameter vector
• Let the 𝐽 x 𝐾 matrix of derivatives of 𝐶 𝛽 be
𝐺=
𝜕𝐶(𝛽)
𝜕𝛽′
• Estimate of the asymptotic covariance matrix is
𝐸𝑠𝑡. 𝐴𝑠𝑦. 𝑉𝑎𝑟. [𝐶 𝛽 ] = 𝐺 (𝑣𝑎𝑟 𝛽 )𝐺’
• Form the Wald-statistic
• Reject the null if the statistic is significantly large
Wald-statistic properties
• For example, consider the restriction, 𝐶 𝛽 = q, which could be
specified by
𝛽1 𝛽2 = 1 or 𝛽1 𝛽2 − 1 = 0
• If 𝑛 is large, both formulations will result in the same statistic
• If the Wald-statistics are statistically different it could affect the result
of the test
Note:
• See the MATLAB Example.
Thank you!