frontier production models - a survey

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Frontier production models A survey

Abstract.

This work aims to collect and synthesize several papers related to efficiency and technological

change, being these two elements constituents of the Total Factor Productivity. We will focus

specially on the agricultural sector in order to give a complete landscape of the knowledge frontier

around the Agricultural issue.

This work is divided in two main parts, efficiency accounting and estimation and by the other

hand, technical change.


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Contents

Total Factor Productivity ..................................................................................................................... 3

Bibliography ...................................................................................................................................... 24


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Total Factor Productivity

TFP is defined as a variable which accounts for the effects on the output not caused by the inputs.

It is composed by two sub-sections, namely Technology Growth and Technical Efficiency. These

two subjects have always been a matter of discussion by economists.

Technology Growth

Technology growth has been subject to analysis at the very beginning of the economical thinking.

We can find references in the works of Adam Smith, Karl Marx, William Petty, among others.

For Adam Smith, technology is the main factor by which the productivity of labor can be improved.

On the Wealth of Nations, Smith states that the Wealth derives from the level of technology use

and both variables are dependent of the division of labor. However this was not Smiths main

concern but the division of labor that triggers productivity improvements. As market grows, the

same thing happens to division of labor. This increased division leads improvements in the

technology level, which derives greater incomes that re-starts the process. (Smith, 1776)

By the other hand, Karl Marx states that the technology change is an important factor that allows

the profit rate to counter-arrest its natural decay rate. By increasing the exploitation of labor and

the intensity of man-hours, new technologies allow capitalists to reduce the wage costs (being

labor the source of wealth) by reducing the time needed for a particular process. By the use of

technology, the profit rate can regain a positive trend as it reduces and enhances the time devoted

on a productive process. (Marx, 1830).

So we can see that Technology Growth and Technical efficiency are matters of concern for the

economics science since its early stages.

(Complete with the rest)

The seminal works by which growth accounting gained interest were (Harrod, 1939), (Domar,

1946), (Solow, 1956) and (Kuznets, 1971), works that falls into the exogenous growth theories.

Most recently, endogenous growth theories were developed, (Helpman & Grossman, 1994).


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Growth accounting theories intended to explain the reasons behind some stylized facts around

economic growth. The breakthrough development was written by (Solow, 1956) who proposed a

model of decomposed contributions to output according to constant returns to scale production

function with capital and labor as inputs.

The model is as follows:

Letbe a Cobb-Douglas function.1.

Also, let the function to show constants returns to scale:

2.

Which is the same as:

3. With Eq. 3 and Eq. 1 we have:

4. Where, K and L are capital and labor respectively, is the proportion of capital used. And A is a

function of the rest of productive inputs such as technology or productive efficiency, which as

(Abramovitz, 1956) once stated, is the measure of our ignorance. This is a symptom of the fact

that at that moment, the reasons behind productivity increases were not fully understood. (Dale

W. Jorgenson, 1987).

The econometric estimation can be made by applying logarithms in order to obtain a lineal

equation to deal with.

5. The estimation is obtained by using OLS, COLS or Maximum-Likelihood methods by assuming that:


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Then:

6. While

measures the mean efficiency level across firms and over time,

is the time- and

producer-specific deviation from that mean, which can then be further decomposed into an

observable (or at least predictable) and unobservable component:

7. 8.

Where represents firm-level productivity and is a white noise source representing unexpected

deviations from the mean due to measurement error, unexpected delays or other external

circumstances. In that sense, the residuals are taken as an indirect measure of technological

change and productivity, and the estimation can be obtained:

9. Growth estimation and accounting theories waned during the 70s as the persistent instability of

the decade drove the economics research interest on to the determinants of cycle fluctuations.

Within the debate on convergence in the early 90s growth theories revived. The inconsistency

between the rates of convergence was highlighted and Romer deduces that in order to show

consistency with reality, the Solow model should account twice as large the current capital

accumulation considered. In that sense, increasing returns to scale and spillovers of the returns to

private investment to the rest of the economy were proposed as a way to overcome the issue.

(Romer, 1986)

Mankiw, (Mankiw, et al., 1992) refloated the Solow model and Romers observations were

accounted. The results achieved high performance in terms of consistency with reality. Basically,

Mankiw et al expands the Solow model by assuming that total investment is an aggregation of two

components, investments in Human Capital and investments in Physical Capital. In that sense, eq.

5 is rewritten:

10. Allowing these differences, roughly 78% of the GDP per capita variations were attributed to the

considered variables. (Mankiw, et al., 1992)


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In the same line, authors as (Hayami & Ruttan, 1970), (Hayami & Ruttan, 1985), adapt the Solow

model in order to study Agricultural Production. By estimating a Cobb-Douglas function affected

by a vector of inputs and assuming the classical residual distribution, the technical change and

efficiency is observed as part of the residual. Total Factor Productivity is then estimated indirectly,

as discussed before.

In spite of being a popular technique in most researches, TFP estimation is subject to critics. By

estimating indirectly the TFP, the first question that arises is how much of this residual is linked to

TFP growth and how much of the remains are attributed to undesirable effects such as accounting

errors, variable omission, deviations from aggregate variables and bad model specification.

(Hulten 2001)

The next section enumerates issues around the TFP estimation.

Methodological issues

Using OLS leads to biased productivity estimates, caused by the endogeneity of input choices and

selection bias. Furthermore, the absence of perfect competition in input/output markets, together

with an omitted variable bias will arise in standard TFP estimation if data on physical inputs and

output and their corresponding firm-level prices are unavailable. Also if firms produce multiple

products, potentially differing in their production technology; failure to estimate the production

function at the appropriate product level, rather than at the firm level, will also introduce a bias in

standard TFP measures (Beveren, 2007).

Selection bias:

Usually, exit and entry of firms is accounted for TFP estimation by constructing a balanced panel.

This mean omitting all firms that enter or exit in the same period (Olley, 1996 as cited in Beveren,

2007). Nevertheless, growth and exit of firms is motivated largely by productivity differences at

the firm level, this concept can be found in (Jovanovic, 1982), (Hopenhayn, 1992), (Farias &

Ruano, 2005) and (Dunne & Samuelson, 1988)

Given that low productivity firms have a stronger tendency to exit, the omission of entering or

exiting companies will likely lead to biased results. As more efficient companies are accounted, the

efficiency residual tends to zero (as efficiency rises) but there is a strong correlation between

capital input and lower inefficiencies. This is because firms with a higher capital supply will be able


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to withstand low productivity without exiting. If firms have some knowledge about their

productivity level prior to their exit, this generate a negative correlation between and K, causingthe capital coefficient to be biased downwards in a balanced sample (Beveren, 2007). This issue is

considered in the works of (Wedervang, 1965) and taken specifically into account by (Olley &

Pakes, 1996)

Simultaneity bias:

Simultaneity bias is defined as the correlation between the level of inputs chosen and unobserved

productivity shocks (De Loecker, 2007 as cited by Beveren, (2007)).

Even though (8) can be estimated with OLS, the method requires the inputs to be exogenous (i.e.

independent of the firms efficiency level). The early work ofMarschak & Jr. Andrews (1944) takes

account of the fact that inputs in the production function are not independently chosen but rather

determined by the characteristic of the firm, including its efficiency.

If the firm has knowledge of its efficiency, endogeneity arises since input quantities will partly be

determined by prior beliefs about productivity. If there is serial correlation in , a positiveproductivity shock will lead to increased variable input usage, introducing an upward bias in the

input coefficients for labor and materials

Omitted output price bias

Typically firm-level prices are unavailable to the researcher, in their place industry level prices

indices are applied to deflate firm level sales. But, if firm level price variation is correlated with

input choices this will bias input coefficients. Assuming that inputs and outputs are positively

correlated and output and price are negatively correlated, the correlation between inputs and firm

level prices will be negative, resulting in a negative bias for the coefficients on labor and materials

(De Loecker, 2007)

These issues can be avoided by using quantities of output rather than deflated sales. But since it

requires information on actual firm level prices this method is rare. This can be found in works as

(Dunne & Roberts, 1992), (Eslava, et al., 2004), (Foster & Syverson, 2008), (Jaumandreu, 2004),

(Mairesse & Jaumandreu, 2005).


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Another approach is to introduce the demand as output into the system and solve at firm level

prices as suggested on (Griliches & Klette, 1996)

Omitted input price bias

In the presence of imperfect competition in input markets, input prices are likely to be firm-

specific. However, since input prices (like output prices) are typically unavailable, quantities of

inputs are usually proxied by deflated values of inputs for capital and materials (the amount of

labor used tends to be available in annual accounts data commonly used to estimate production

function relationships). Assuming that quantities of output are given, this leads to the following

relationship, where is the deflated value of capital, and are firm-level and industry-levelprices:

Then it is clear than in the presence of unobserved firm-level input price differences, coefficients

on

will be biased. (Beveren, 2007)

In response to this, whilst works as (De Loecker, 2007) argues that the bias issue can be sorted if

the imperfect output markets are treated explicitly, as higher input prices are reflected in higher

output prices which in turn depends on the relevant firm-level mark up. Nevertheless, even in a

competitive market, adjustment costs will lead to differences in the price of the input index across

firms induced by differences in current levels of the quasi-fixed factors, as capital. (Beveren, 2007)

Endogeneity of the product mix

When we consider the case were firms produce multiple products for the same industry and these

products differ in the production technology or in the demand they face, this will lead to biased

TFP estimates, since the production function assumes identical production techniques and final

demand (Beveren, 2007).


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Consistent TFP estimation in the context of multi-product firms requires information which is not

always available for research (product-level output, inputs, as well as prices). In that sense, a

number of partial solutions are proposed. On the absence of information about inputs/outputs at

product level, it is possible to sort firms into groups making a single product. On the other hand if

the number and type of products are known, the parameters of the production technology can

vary allowing consistent estimates.

Productivity sources

Productivity and enhancement sources are found on Technology and Knowledge transfer, being

education and knowledge diffusion considered as part of the inputs. These ideas can be found on

works such as (Beal, 1978), (D. & Lau, 1982), (Griliches, 1964), (Vanzetti & Bessell, 1974), and in a

more general sense, (Schultz, 1963).

A more recent work (Islam, 1995) extended this model allowing differences in productivity

between countries. By doing this, the Solow model is able to capture endogenous accumulation of

Human Capital without the need of Human Capital accounting. The author conclusion is that

Human Capitals contribution to changes in growth might not be as evident due to its slow change

rate: While physical capital fully adjusts in a decade, human capital may require a century to

respond to changes in educational policies (Fuhrer & Little, 1996).

In spite of the academic research work devoted to find strong evidences between education and

productivity, most of the empirical work fails on taking account the fact that agricultural

technologies changes over time, is because of this that Empirical analysis assuming homogenous

technology thus may obscure the true contribution of education to agricultural productivity

(Huang & Luh, 2009).

Using a switch regression analysis1, (Huang & Luh, 2009) examines the effect of education on TFP

using a combination of a Malmquist Index (MI) and a Switching Regression Model (SRM). By the

estimation of the MI, two regimes can be chosen based on its result. Specifically, the proposedmodel is:

11. * +1

For an example of switch regression analysis see appendix.


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The distance function in the numerator, , measures the maximal proportionalchange in output required to make feasible in relation to the technology at t . In thedenominator, the distance function measures the reciprocal of the maximumproportional expansion of the output vector

given

The second term in the bracket is similarly

defined as the Malmquist productivity index with technology in period t +1 as the reference

technology. (Huang & Luh, 2009).

The Malmquist productivity change index can be calculated through the linear programming

approach2. The basic idea in the nonparametric programming technique is to construct a world or

best-practice frontier from the data in the sample, and then compares individual countries to the

frontier (Huang & Luh, 2009). This value is used here as a separation index for the switching

regression analysis, it can be interpreted as a technical change component, which measures the

shift in the frontier over time. This is the amount by which the frontier shifts at each countrys

observed input, and it will be used to separate the time span into two regimes, progress and

stagnation (Ibid.)

Hence, we define TCC as:

2for an introduction to non-parametric methods, please refer to (Racine, 2008)


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And our switching model:

(Regime 1)

(Regime 2)

The switching mechanism is defined by:

, The variables are defined as:

: Used to quantify the direct influence of education through the percentage of secondaryschool enrolment

: Disembodied technological change: Country-specific agricultural farming characteristics land/labor ratio, is calculated asarable land per unit of agricultural labor.

What is interest on this model is its ability to account some particularities which cannot be

assessed by using traditional methods (i.e. Single regimes models). By taking account the Farmers

ability to deal with disequilibria induced by technical change, a more efficient version to testing

the hypothesis that education plays a key role in agricultural development arises (Huang & Luh,

2009)

As education, investment has been pointed as a determinant of productivity. Investment can be

private or public and can also be differentiated in sub-forms, such as Research and Development

Expenditure (a term a bit confusing as R&D expenditure shows a return in the form of new

products/processes), or Infrastructure Investment.


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Technical efficiency

Technical efficiency was first addressed by Farrell. The author defines the efficiency of a firm as

its success in producing as large as possible an output from a given set of inputs (Farrell, 1957).

As an initial example, Farrell defines a firm employing two factors of production to produce asingle product, under conditions of constants returns to scale. Also, the production function is

assumed to be known.

Under the previous assumptions, an isoquant diagram can be used to represent graphically the

concept of technical efficiency. In figure 1, P represents the inputs of the two factors, per unit of

input used by the firm. The isoquant SS represents the different combinations of the two factors

that a perfectly efficient firm uses to produce a unit of output. The point Q represents an efficient

firm using the factors in the same ratio as P. Then we can see that the same amount can be

produced using only a fraction 0Q/0P than P. Therefore, the ratio 0Q/0P can be defined as the

technical efficiency of the firm (Farrell, 1957).

Also, in the view of the prices, it is necessary to measure the best proportion of factors that the

firm should use. As seen in Figure 1, AA has the same slope as the ratio of prices of the two

factors, therefore Q represents the optimal combination of factors that allows the maximum

Technical and Alocative efficiency.

Fig. 1


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The estimation of the production function is done by constructing a hypothetical firm as a

weighted average of two observed firms. The conditions for the curve are that its slope is never

positive and there is no observed point between the curve and the origin (see Fig 2)

Fig. 2

The generalization for the case of several outputs/inputs is pretty straight-forward and involves

the use of input/output vectors and their corresponding estimated parameters. This can be solved

by using programming methods.

A difficult faced by Farrell was the case of production function showing scale economies. Onestimating the production frontier, the relaxation of the constant return to scale assumption

implies some technical difficulties. As the output grows faster than the use of inputs, the efficiency

prediction might be too optimistic in case of diseconomies of scale and vice versa. As first

approach, Farrell proposes to apply the method by creating groups of observations having similar

levels of output.

The same happens when endowments differences are accounted. This is what Farrell enounces

as quasi-factors, for example the thickness of seam in coal mining. The author proposes the same

solution as in the case of (dis)economies of scale (i.e. divide the observations into groups

homogeneous in the quasi-factor (Farrell, 1957)).

Furthermore, the interpretation of efficiency is discussed. Farrell splits the efficiency concept into

two elements, namely Technical Efficiency and Allocative Efficiency. While the first specifies the


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rate of success of a firm on producing as much output possible given a set of inputs, the second

takes account of the different input combinations and having in mind the prices, specifies the

efficiency in terms of factor allocation (Seen in Fig. 1 as point Q). It is necessary to point that price

efficiency depends on several factors, being affected by the slope of SS and AA. Introducing new

observations probably will change the curvature of SS. Additionally, assume a constant slope on

AA implies perfectly elastic supply of each factor, whilst it is expected to change as one input

becomes more and more used. This in the end will result on underestimate the price efficiency of

the firm.

Farrells isoquant was the beginning of the frontier production functions later developed and while

it was used to evaluate the performance of production organizations, frontier production and cost

functions can be used as efficiency standards and provide information on the features of the best

practice technologies as well. However these measures are different from those of Farrell, where

his indexes hold output constant and focus on differences in production levels. Starting with (Chu,

1968) and (Afriat, 1972) frontier production and cost functions treat the technical parameters of

the production function and the efficiency parameters simultaneously (Kopp, 1981).

Frontier production functions

Whilst Farrells intention was to evaluate the performance of production organizations, frontier

production and cost functions can serve as efficiency standards and provide information on the

features of the best practice technology as well (Kopp, 1981). Frontier production and cost

functions can be used to evaluate efficiency by taking account of deviations between observed

values and the expected values given the empirical frontier function. This characteristic leads to

different estimation settings and has implications on the econometric interpretation of the values.

Let be a production function, where Q is an output vector, X is a matrix ofobservation, a vector of unknown production parameters and a vector of random

disturbances.

The traditional average production function estimation would specify as required by the OLS

conditions on random disturbances (independent distribution and n~(0,2)). On the other hand,

frontier estimators would expect to be non-zero, reflecting the existence of inefficiency. The

assumptions made around the disturbance term allow us to classify models on three groups:


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i. Unspecified random shock expected to be less than or equal to zero.ii. Random disturbance which follows a one-sided distribution.iii. Composite noise with both symmetric and one-sided distributions.

Given that models differ basically on their disturbance term due to their underlying assumptions,these are reflected on the way productive efficiency is accounted (via the disturbance term).

(Aigner, et al., 1968) and (Timmer, 1971) assumes all variation due to technical efficiency, hence

residuals are restricted to be of one sign. Technical efficiency is specified in these models as the

ratio between observed output and maximum output technically feasible.

(Afriat, 1972) and (Richmond, 1974)


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Total Factor Productivity: Solving estimation issues.

Fixed Effects Estimation

This involves the assumption that in eq. 8 is time invariant and specific to each productive unit,

its estimation is possible using the fixed effects estimator (Pavcnik, 2002), (Levinsohn, 2003)

The estimation equation becomes:

1. Eq. (11) can be estimated in levels (using LSDV, i.e. including firm specific effects) or first/mean

differences. On the assumption that has no time variation, the estimator will be consistent.Under this set up, simultaneity and selection bias is overcome. The later by eliminating

entries/exits considerations and the former by assuming that has no time variation (i.e. the firmhas no prior nor need to have - knowledge of at the time input decisions are made)In spite of the attractiveness of this estimator, empiric results are unsatisfactory, leading to

underestimation of capital coefficients. Furthermore, (Olley & Pakes, 1996) finds large differences

on the estimation of balanced vs. unbalanced sample sets using this method, suggesting that the

assumptions underlying this model are invalid (Beveren, 2007)

The time-invariant nature of in the fixed effects model has been relaxed by (Blundell, 1999) inthe context of production functions, by allowing productivity to be decomposed into a fixed effectand an autoregressive AR(1)-component.

Instrumental Variables (IV) and GMM

In order to achieve consistency of coefficients in the production function, independent variables

causing endogeneity (inputs) are instrumented by regressors correlated with them but

uncorrelated with unobserved productivity (our variable of interest).

Three conditions are required. The instrumented variables must be correlated with the

endogenous variables. Secondly, the IV cannot enter the production function directly. Third, IV

cannot be correlated with the error term.


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Assuming perfect competitive markets, input and output prices are the natural choice for IV. But

as in Fixed Effects Estimation, empirical results are not satisfactory. The lack of appropriate

instruments in most data set is evident. Input and output prices are usually not reported.

Additionally, even though IV techniques overcome the simultaneity bias, it does not provide a

solution for selection issues. When input and output prices are taken as IV and entry and exit

decisions are based on these variables, results will remain biased.

Olley-Pakes estimation algorithm

(Olley & Pakes, 1996) Proposes an alternative method using a semi parametric estimator that uses

the firms investment decision to proxy for unobserved productivity shocks. Selection bias is solved

by creating an exit rule. For a deep understanding of this method, it is recommended to refer to

(Olley & Pakes, 1996).

As said before, the traditional way of accounting for entry and exit restricts the analysis to a

balanced panel (i.e. a data set that consists of only those firms that were present during the

whole period considered). As firms exit/entry decisions are subject to their perceptions of their

future productivity (Jovanovic, 1982), (Hopenhayn, 1992), (Farias & Ruano, 2005) and (Dunne &

Samuelson, 1988), the resulting balanced panel will be selected in part on the basis of unobserved

productivity realizations, generating a selection bias in the production function estimates.

By the other hand, the early work of Marschak & Jr. Andrews (1944) takes account of the fact that

inputs in the production function are not independently chosen but rather determined by the

characteristic of the firm, including its efficiency.

If the firm has knowledge of its efficiency, endogeneity arises since input quantities will partly be

determined by prior beliefs about productivity. If there is serial correlation in , a positiveproductivity shock will lead to increased variable input usage, introducing an upward bias in the

input coefficients for labor and materials

Behavioral Framework

To solve the simultaneity problem, the model must specify the information available when input

decisions are made. And in order to overcome the selection bias, an exit rule must be present in


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the model3. This rule is created by assuming that current profits are a function of the Firms owns

state variables, factor prices and a vector which lists the state variables of the other firms active in

the market:

Firm specific state variables:

Market structure:

Factor prices:

Common across firms, following an exogenous First Order Markov process.

At the beginning of each period, the Firm faces three decisions to make. First, to decide whether

or not to continue in operation. If it exits, the Firm receives a sell-off value of dollars and never

reappears again. If it decides to continue, it chooses variable factors (labor) and a level of

investment, which together with the current capital value determine the capital stock at the

beginning of the next period:

1) The index of productivity is known to the firm and evolves over time according an exogenous

Markov process. The distribution ofconditional on all information known at t is determinedby the family of distribution functions:

2) |

The Firm is assumed to maximize the expected discounted value of future net cash flows.

Therefore, entry or exit decisions will depend on the Firms perception of the distribution of future

market structures based on the current information. The future market distribution will be

3For further information on behavioral models, please refer to (Ericson & Pakes, 1995), (Hopenhayn &

Rogerson, 1993), (Jovanovic, 1982) and (Lambson, 1992)


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generated, in turn, by the investment, entry and exit decisions. (Ericson & Pakes, 1995) Prove the

existence of a Markov perfect Nash equilibrium an equilibrium where firms perceptions of the

distribution of future markets structures are consistent with the objective distribution of market

structures that the firms choices generates. Assuming the existence of such equilibrium, the

profit-maximization equation is as follows:

3) { [ |]}Where:

The operator in 3) indicates that the firm is comparing the sell-off value with the expected

discounted returns of staying in business. If the current state variables indicate that keeping on

business is not worthwhile, the firm closes the plant down. If this is not the case, the Firm chooses

an optimal investment level (which cannot be negative). The solution to this control problem

generates an exit rule and an investment demand function. If we define to be zero when theFirm closes down, then the exit rule and investment demand function can be rewritten as:

4) And

5) and are determined as part of the Markov perfect Nash equilibrium behavior. These areindexed by t as they depend on the market structure and prevalent factor prices when decisions

are made.

Estimation

Assuming a Cobb-Douglas production function:

6.


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Where is productivity and is either measurement error (which can be serially correlated) ora shock to productivity which is not predictable, both variables are unobserved. The difference is

that is a state variable in the Firms decision problem and a determinant of both liquidationand input demand decisions, while

is not. (Olley & Pakes, 1996)

We first consider the simultaneity bias caused by the endogeneity of input variables. Endogeneity

arises as input choices are partially determined by the Firms beliefs about . If there is serialcorrelation in , inputs in period t will be positively correlated with it and the OLS procedure willtend to provide upwardly biased estimates of the input coefficients.

Then we consider the self-selection issue induced by plant closings. If we assume that there are

not variable factors, the conditional expectation ofyt(conditional on current inputs, survival and

information available in t-1) includes the term:

7. [| ] As stated in 3) the profit function increases in k, the value function increases and decreases ink. Hence, firms with larger capital stocks can expect larger future returns for any given level of

current productivity, and they will continue in operations at lower productivity realizations. Given

this, the self-selection generated by exit behavior implies that 7) will be increasing in k leading to a

negative bias in the capital coefficient.

Now, we describe the estimation algorithm. Labor is assumed to be the only variable factor (i.e. its

choice can be affected by the currentvalue of). The other inputs and are fixed factorsonly affected by the distribution of conditionalon the information available at t-1 and pastvalues of. In this case, the solution to the optimization problem faced by the firm in 3), resultson equation 5) for investment. Provided that , this equation is strictly increasing in for every(a, k)4. Consequently, for the subset of values for which , 5) can be inverted as:

8) Equation 8) allows us to express the unobservable variable as a function of observables.Substituting 8) into 7):

4For a demonstration on this postulate, please refer to Pakes A. (1994) Dynamic structural models,

problems and prospects part II: Mixed continuous discrete control problems and market interactions

Advances in Econometrics, ed by C. Sims. Cambridge: Cambridge university press.


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9) Where:

10)

The model in 9) is a semiparametric regression which identifies but not the productionfunction coefficients of capital and age, and , as the equation is not able to separate theeffects of capital and age on the investment decision from their effect on output, these are

estimated from the survival probabilities:

11){ | } { }

Where the second equation in 11) follows from 8) and 1). Having solved this, we now

considerate the conditional expectation of yt+1 based on the information available on t and the

survival conditions:

12)[ | ] [| ] ( )Where

( )is the density function:

( ) | | The bias term in 12) is a function of two indices of firm-specific state variables. To control for the

effect of the unobservable on selection a measure of and a measure of the value of whichmakes the firm indifferent from selling off and continue operations must be found. Most models

used to correct for selection have single indices, here we will use two.

Given the density ofconditional on is positive in a region about for every (seethe integer in 12), the selection equation can be inverted to express as a function ofPt and. By conditioning on the selection probability we can condition on the value of one of the twoneeded indices.


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Given that the density ofconditional on is positive in a region about

Levinsohn-petrin estimation algorithm


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