_{1}

Th is paper investigates inference methods to introduce prior information in econometric modelling through stochastic restrictions. The goal is to show that stochastic restrictions method estimator can be asymptotically more efficient than the estimator ignoring prior information and can achieve efficiency if prior information grows faster than the sample information in the asymptotics. The set up includes the nonlinear least squares and indirect inference estimators. The paper proposes a new indirect inference estimator that incorporates stochastic equality constraints on the parameters of interest. Finally, the proposed approach is applied to a macroeconomics model where high efficiency gains are sh own.

One of the ways in which prior information can be modeled is by the use of the stochastic restrictions approach. The rationale is based on the fact that it brings efficiency gains in the estimators, naturally, subject to the quality of the information available. In some cases, prior information is derived from economic theory, and imposes restrictions among parameters that should hold in exact terms. This prior information could be included in the model as a deterministic restriction, and the restricted estimator has smaller variance than the non-restricted estimator. In other cases, prior information derives from previous estimations of similar models or samples. This information could be considered an approximation of the unknown parameter, or a range of values that should contain the parameter with some probability. In this case, deterministic restrictions should not be included in the estimation procedure since the restricted estimator will be biased. If this information is not taken into account despite being valuable, the chance of improving the efficiency of the estimator is wasted. An intermediate solution is to include it with uncertainty. This is the idea behind the stochastic restrictions approach and it is shown to bring efficiency gains (as shown in [

Despite the finite sample efficiency gains of the Theil-Goldberger approach, this result cannot be extended to the asymptotic distributions of the constrained estimator, since the efficiency gain vanishes as sample size increases. This result is proved although not useful for empirical research (see [

The main contribution of this paper is the description of a simulation-based estimator in which prior information is taken into account through the stochastic restrictions approach, also, under the same type of assumptions already introduced in the first objective. Simulation-based methods, as the method of simulated moments [

The structure of the paper is the following: Section 2 provides the motivation for the assumption that supports the main results of this paper, also, I discuss the derived efficiency gains for a traditional estimator in finite sample and asymptotic terms. Section 3 describes the method to combine prior information into the indirect inference criteria through the stochastic restriction approach. Also, asymptotic properties are provided. Section 4 focuses on a macroeconometrics example and numerical evaluation of the efficiency gains of the suggested method, and Section 5 concludes.

The first result to be shown in this paper is that stochastic restrictions yield asymptotic efficiency gains under specific assumptions. Previously, In this section, I provide the definition of stochastic restriction and describe how it behave in asymptotic terms in a standard framework. I show that in order to obtain efficiency gains derived from the introduction of stochastic restrictions it is needed to assume a particular behavior of the prior information in the asymptotic setup. This particular assumption is also motivated in this section.

Consider a general linear model,

which is called a stochastic restriction. In the above equation r is a

Let us show how a stochastic restriction could be defined from prior information in a very simple model formed by a Cobb-Douglas production function

One of the key element of this paper lies on the particular assumption I make about the asymptotics of

This kind of assumption might be considered too strong and, as mentioned in [^{1}. Consequently, there is no need to make such “realistic” assumption … the quality of the approximation is the only criterion for justifiability”. The parameter sequence I choose, as mentioned, is specified on the variance of the stochastic restriction, and using [

Finally, it is presented an additional argument to motivate the main assumption considered. In a more specific context, the key assumption allows to blend prior and sample information when estimators based on simulation must be used. This is the case of models that generate high nonlinearities in the traditional criterion, making standard methods useless. Generally, the estimators obtained by simulations, despite the fact that are the only solution to estimate some families of models, show high variance, and hence, efficiency gains would be welcomed. The key assumption allows the I.I. procedure becoming a more efficient procedure if stochastic restrictions are correct.

Efficiency GainsIn this section, I discuss the relevance of taking into account prior information in the estimation of a general nonlinear model. First I remind the properties of the nonlinear least squares estimator (

The purpose of this discussion is to establish formally the setting in which the stochastic restrictions are relevant to explain efficiency gains in the context of a traditional method. This formalization is intended to enhance the understanding of the technical role played by the assumption into

We start our discussion by considering a general nonlinear model given by the following equation

where

exogenous variables. If

to be considered―see, for instance, [

mator. This assumptions are, in vague terms, the existence and continuity of

as T goes to infinity, of the second order derivative of

Now I consider a set of q stochastic restrictions on^{2}. Following the Theil-Goldberger approach, the resulting model in which Equation (2) and stochastic restrictions are considered, can be expressed as:

Since, in general,

where

Some additional notation should be introduced. Let

Least Squares under Stochastic Restriction (SR) estimator of

Our purpose is to compare the asymptotic variance covariance matrix given in (3) and (6), first in the context of the standard asymptotic theory, and also in a general alternative context to be defined, based on the structure of the variance of the error term v. We start first by the standard asymptotic setting where the following result is obtained, already provided by [

Proposition 1. Under SAA, the SR and NLS estimators have the same asymptotic variance covariance matrix.

The proof is immediate. Since

Hence, ^{3}.

Proposition 1 shows that stochastic restrictions bring no asymptotic efficiency gains. The irrelevance of the stochastic restrictions is not a satisfactory result for empirical purposes, where the asymptotic distribution has to be used to approximate the variance of the estimator, especially when the sample size is small. The question that arises is whether or not it would be possible to find a theoretical framework to keep the relevance of the stochastic restrictions in asymptotic terms, as stated in [

The new context is based on the idea that prior information about parameters comes from previous experience. Moreover, experience derives from observations that are taken from a sample of size

Assumption 01 (A01). The variance of v, the error term of the stochastic restriction, is

This assumption states that the quality of the prior information increases asymptotically with^{4}. It should be noted that A01 implies that the term r in the stochastic restriction is also a random term and hence it depends on

The asymptotics is analyzed as T and

Assumption 02 (A02).

Assumption 03 (A03).

Assumption 04 (A04).

The purpose of (A02) is to maintain equal weights of prior information and sample information in the limit. Assumption (A03) states that sample information increases more rapidly than prior information, while (A04) set the opposite. We will show that for the three cases, the variance of the SR vary between an inferior bound, given by the variance of the deterministically restricted estimator (I will simply call this as the restricted estimator and denoted it as

Proposition 2. Under SAA, (A01) and (A02),

This result shows that stochastic restrictions brings asymptotic efficiency gains with respect to the NLS esti- mator when sample and prior information increases at the same rate. In other words, Proposition 2 recovers the Theil-Goldberger contribution for asymptotic distributions, and for the resulting approximated finite sample distributions so derived.

Proposition 3. Under SAA, (A01) and (A03),

This result shows that stochastic constraints do not increase efficiency when sample information increases more rapidly than prior information. In other words, Proposition 3 shows the standard asymptotic conclusion of Proposition 1 as a particular case of the general analysis described by assumption A01.

Proposition 4. Under SAA, (A01) and (A04),

This result shows that when prior information increases more rapidly than sample information, stochastic constraints increase efficiency to the level of the restricted estimator

Finally, in the following proposition I show a concluding result for a varying

Proposition 5. Under (A01), as

1)

2)

3)

We have established a general setting in which several goals are covered. First I have stated a unique analytical context to explain restricted and non-restricted estimators, in the general terms of the stochastic restrictions approach. In this context, restricted and non-restricted estimators are particular cases of^{5}.

The indirect inference method is a simulation-based moment matching estimation procedure. The general idea is to match the moments of the auxiliary model from the simulated data to observed data to obtain the estimates of the structural parameters. The method of Indirect Inference (I.I.) of [

First, I define the Indirect Inference under Stochastic Restrictions (IIR) estimation method and provide its distribution. Then, based on the approach introduced in Section 2, I show that the IIR estimator is more efficient than the I.I. method, provided that the stochastic restrictions are asymptotically correct.

In the I.I. approach it is considered a p-dimension vector of parameters

Some facts have to be pointed out in order to understand the principle of the I.I. estimation. It is assumed that it is not feasible to estimate M by mean of a conventional method, due to its complexity or intractability of a conventional criterion for that model. On the other hand, ^{6}. As of the optimization on

full rank on a neighborhood of

where

below. Under regular assumptions about the auxiliary criterion

where

and

Appendix 2.

The matrix

Since

variance of

We now consider the existence of prior information on the parameters of interest

Function

It is necessary to introduce some additional notation to define the proposed estimation method. Let

tively.

Definition The Indirect Inference under Stochastic Restriction (IIR) estimator of

where

Some additional assumptions are in order to derive the asymptotic behavior of the IIR estimator.

(A1) - (A7). Are the regular conditions needed to obtain the asymptotic distributions if the I.I. estimator, shown in Appendix 2.

(A8)

(A9)

(A10)

Assumption (A9) describes the asymptotic properties of the stochastic restrictions, and it leads to the appro- ximate distribution

and hence similar to assumption (A01) introduced in Section 3. Again, the rationale behind (A9) is the intention to maintain a constant relative weight between the sample and prior information asymptotically. The relevance of this assumption lies in the fact, already discussed, that under these hypotheses, the approximate distribution for small sample size of the resulting estimator is closer to the observed distribution of the estimator. Note that (A9) implies consistency of the random variable^{7}. The asymptotic properties of the IIR estimator are derived next.

Proposition 6 Under assumptions (A1) to (A10),

where

This result is proved in Appendix 2.

For the optimal matrix

where

Proposition 7 Under assumptions (A1) to (A10)

To proof this result, I compare Equation (7) and Equation (9). The difference

is a negative definite matrix, since

is a positive definite matrix.

This section conducts a set of empirical estimations to assess the performance, in terms of bias and efficiency, of the estimation method described in Section 4 compared to the I.I. method.

The model of interest is given by a production function and the perpetual inventory method equation for the capital stock, K, which depends on the depreciation rate,

To go further into the economic motivation of the model, note that K is one of the basic economic aggregates, and following the definition provided by the perpetual inventory method, it is given by:

where I is investment and d the depreciation rate, which measures the loss in value of the existing capital stock as it ages. Since d is an unknown parameter, K is not observable and in practice it is usually measured by accounting techniques, which provides not satisfactory figures since, for instance, technological shocks have not effects on the actual value of the net capital stock. One solution to measure the capital stock is by mean of the simultaneous estimation of d jointly with the parameters of a production function,

The theoretical model of interest is given by a Cobb-Douglas production function, and assuming constant returns to scale becomes:

where y, l and k are production, labour and capital stock in logs respectively,

stock and it is assumed that

In the above equation,

of

dent of

Besides, the model introduces a prior value

where

respectively) and v is the error term capturing the uncertainty about

Case | I. Constant | II. Dummy variable | III. Growing Rate of Y |
---|---|---|---|

Stochastic restriction |

The empirical model of interest is formed by the following equations:

where data requires

gressive structure ^{8}. It should be noted that data allows for a structural change for the intercept, captured be

possible dates. For the Case III equation, the rate explanatory variable

role of the intensiveness in using the capital stock in the depreciation pattern. Finally, it is assumed that

The parameter vector of model (14) is

which is estimated by IIR using data of the variables y, l, I,

The auxiliary criterion is maximum likelihood and the auxiliary model for the IIR estimation is exactly the same model considered in [

where

random error is considered to follow an AR(1), capturing the total factor productivity dynamics and yielding more accurate estimates. Finally, the parameter vector of the auxiliary model (15) is

The motivation for structure of the auxiliary model relies on the fact that it is a more simple model, since no random term is considered in the equation of the variable rate of depreciation, and, on the other hand, it is a more general model, since no constant returns to scale are imposed in the production function. Very little can be said in priors grounds about the adequacy of one specific model to be the best auxiliary model for I.I. estimating. Nevertheless, it is in general admitted that the model should be similar, and if possible, more general. Both of this characteristics are considered in the selection of the model considered, which is also supported by the empirical results.

As defined in the previous section, the IIR estimator of

being

where

where

involved in the definition, that is

The ratio

The point estimate of 0.3 for

Columns 6 and 7 give the results for Case III, and the coefficient

In a more general setting,

Case | I: | II: | III: | |||
---|---|---|---|---|---|---|

Method | IIR | I.I. | IIR | I.I. | IIR | I.I. |

5.035 | 5.022 | 5.199 | 5.116 | 5.166 | 5.167 | |

(4.83) | (4.03) | (3.14) | (3.40) | (3.73) | (3.21) | |

0.040 | 0.040 | 0.0403 | 0.039 | 0.041 | 0.041 | |

(1.69) | (1.45) | (1.97) | (1.76) | (1.73) | (1.58) | |

0.291 | 0.300 | 0.2975 | 0.298 | 0.297 | 0.291 | |

(5.04) | (4.30) | (6.22) | (6.22) | (1.72) | (1.20) | |

0.200 | 0.200 | 0.201 | 0.204 | 0.206 | 0.206 | |

(1.75) | (1.62) | (1.50) | (1.45) | (2.42) | (2.20) | |

1.1 × 10^{−}^{4} | 1.1 × 10^{−4} | 1.1 × 10^{−4} | 1.1 × 10^{−4} | 1.0 × 10^{−4} | 1.0 × 10^{−4} | |

(9.3) | (9.3) | (9.5) | (9.5) | (10.01) | (9.1) | |

0.045 | 0.040 | 0.065 | 0.071 | 0.067 | 0.070 | |

(1.94) | (1.68) | (2.65) | (1.50) | (2.05) | (1.70) | |

- | - | 0.010 | 0.010 | 0.009 | 0.008 | |

(2.53) | (2.47) | (2.18) | (1.38) | |||

0.017 | 0.016 | 0.016 | 0.017 | 0.015 | 0.015 | |

(0.42) | (0.03) | (0.15) | (0.15) | (0.55) | (0.35) |

^{1}t-values in brackets.

with very little difference from the baseline model estimates which contains no stochastic depreciation rate. This result allows for confidence in terms of bias and adequacy of the simulation-based methods for the estimation of this specific model, although not significant differences are found for the estimates of the parameters underlying the variable depreciation rate.

Second, IIR is more efficient than I.I., which is shown for the parameter for which prior information is available. In fact, efficiency losses are small provided that I use conservative choices for the variance of the stochastic restriction. Alternative estimations were conducted for different quality levels of the prior information, confirming that efficiency losses are inversely related to the quality of the prior information.

Third, in the implementation of the IIR method, convergence is achieved faster than for the I.I. estimation, which shows that the proposed methodology is a practical way to mix prior and sample information in a simulation-based estimation method. On the other hand, preliminary results suggest that by reducing the number of simulations (say, to 50), it will be possible to reduce the computation time of IIR without adversely affecting its finite sample properties.

This paper formalizes some intuitions about the role of prior information on asymptotic rules of inference. In particular, the natural idea that despite prior information is asymptotically irrelevant when modeled through stochastic restrictions, this theoretical result may not avoid using accurate prior information for empirical purposes. Nevertheless, so far there is no any contribution in the literature providing ground for it.

Asymptotic theory is a tool that provides approximate figures for the mean and the variance-covariance matrix of estimators that in general may have an empirical interest, that is, may be one of the few practical solutions to estimate a model of interest. Nevertheless, if prior information is irrelevant in asymptotic terms, it will be so in the derived finite sample approximation of the variance of such estimator. This result of course is not helpful and leads to discard any use of prior information even knowing that prior information in general may be relevant if accurate-in terms of efficiency. This paper is intended to provide an insight in the previous discussion in the sense that if prior information is proved to be asymptotically relevant, then it will also be for the finite sample approximation and thus will bring efficiency gains on empirical ground. This previous discussion is the motivation of this paper and the solution I provide may be understood as a contribution oriented to enhance the usefulness of any estimator as in asymptotic terms there is no room for using prior information in the form of stochastic restrictions.

On the other hand it is worth it to recall the large variance of the I.I. estimator (as well as of others simulation based estimators). This additional setup provides specific motivation to face the challenge of providing theoretical ground for the asymptotic efficiency gains due to stochastic restrictions.

The main contribution, which is the formulation of a new estimator (the IIR estimator), more efficient than the baseline estimator is achieved through the introduction of one specific assumption, which in short is that prior information increases with sample size. This idea, the cornerstone of the suggested approach, is intended to be taken as a potential contribution for the large family of simulation based estimators in the sense that they are now allowed to mix sample and prior information to achieve efficiency gains.

As expected, this discussion is open for future research as empirical results that may be found for testing this insight, may support it or not.

José A. Hernández, (2016) On the Asymptotics of Stochastic Restrictions. Theoretical Economics Letters,06,707-725. doi: 10.4236/tel.2016.64075

We set the following assumptions to prove Propositions 2 to 5.

(A01).

(A02).

(A03).

(A04).

Proposition 2. Under SAA, (A01) and (A02),

Proof. Under (A01), by construction, the asymptotic distribution of

From (A02),

and

is a definite negative matrix, and

what means that efficiency gains are also extended to finite sample distributions.

Proposition 3. Under SAA, (A01) and (A03),

Proof. To prove this proposition I use the general form of the Sherman-Morrison-Woodbury formula (see [

where A and C are

From the distribution given in (16) and taking into account Equation (17), I can rewrite

since by (A03)

Proposition 4. Under SAA, (A01), and (A04),

estimator of the model.

Proof. From the rewritten equation of

Since (A04) states that

and, by substituting the above equation into the equation of

which is easily checked to be the asymptotic variance covariance matrix of the NLS estimator of the model

which is the restricted model.

Proposition 5. Under (A01), and as

1)

2)

3)

Proof. Taking limits in the term where

and going back to the Equation (18), I have

Since

From (19),

then, by substituting the above results into (18), I obtain

Here I develop similar proofs to the used on the asymptotic properties of the I.I. estimator. To show the asymptotic distribution of IIR estimator I need several regularity conditions, as for the I.I. distribution. The most important are

A1) The general auxiliary criterion function

A2) This limit function has a unique maximum with respect to

A3)

A4) The solution of the asymptotic first order condition

A5)

A6)

A7)

A8)

A9)

A10)

Let us first prove the consistency of the IIR estimator. Under assumptions (A1) to (A4), following [

proved that the intermediate estimators

and

Let us now find the asymptotic distribution of

of

and

The asymptotic expansion of

An expansion around the limit value

since

the limit, and calling

From (21), (20), I get

and using (A6), (A7),

where

Finally, using assumptions (A8), (A9) and (A10):

where

The optimal

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