3.1 Production technology and investment

The real side of the firm is characterized by a production technology that uses capital $k$ and labor $n$ . Revenue per period is given by $y = zk^{\alpha }n^{1-\alpha }$ , in which $\alpha$ and $1-\alpha$ are the production elasticity of capital and labor, respectively, and $z$ is a productivity shock. Since this paper aims to investigate how any movement in interest rate could affect the firm’s decisions through the working capital channel, I normalize $z$ to be equal to 1 in the model. Furthermore, Investment, $I$ , is defined as

(3) \begin{equation} I \equiv k' - (1-\delta )k, \end{equation}

in which $\delta$ is the capital depreciation rate, $0\lt \delta \lt 1$ , $k'$ is the next period capital stock, and $k$ is the current capital stock. I assume that investment is partially irreversible to capture the fact that the firm’s investment is lumpy (e.g. Doms and Dunne, Reference Doms and Dunne1998; Caballero, Reference Gali and Gertler1999). Following Michaels et al. (Reference Michaels, Page and Whited2019), I normalize the price of investment goods to one, and the price of selling capital (negative investment) is expressed by $\theta _{i} \in (0,1)$ . Irreversible investment suggests that the firm sells capital at a lower price than it paid to buy it. As a result, the cost of investment is $C(I)$ defined as

(4) \begin{equation} C(I) \equiv I\cdot 1_{[I\geq 0]} + \theta _i I\cdot 1_{[I\lt 0]} \end{equation}

3.2 Working capital loans

It is well-known in corporate finance literature that a firm needs to cover the cash flow mismatch between the payments made at the beginning of the period and the realization of revenues (Mahmoudzadeh et al. Reference Mahmoudzadeh, Nili and Nili2018; Michaels et al. Reference Michaels, Page and Whited2019). These funds in advance needed by the firm are known as working capital. I model this fact assuming that a firm needs to finance a fraction $\phi$ of its total variable cost in advance. I assume that total variable cost is generated by labor input $wn$ , in which $w$ is the real wage per hour and $n$ represents the number of working hours. As a result, $\phi wn$ represents the working capital needed by the firm, which would be financed by working capital loans at the beginning of the period. Considering that $R$ is the gross interest rate for working capital loans, the total variable cost faced by the firm is $R \times (\phi wn)$ , which is paid at the end of the period. Furthermore, the gross interest rate $R$ follows a discrete Markov process.Footnote ⁵

(5) \begin{equation} R' = R_{ss}(1-\rho ) + \rho R + \varepsilon ', \quad \varepsilon \sim N\left(0,\sigma _{\varepsilon }^2\right), \end{equation}

in which $R_{ss}$ is the steady-state value of $R$ , $\rho$ is the persistent parameter, and $\varepsilon$ is the interest rate shock. The innovation $\varepsilon$ has a normal distribution with mean zero and variance $\sigma _{\varepsilon }^2$ . How $R$ is modeled allows us to consider the real mean value of $R$ , which is obtained from the data. The set of possible values for $R$ is bounded since $R$ cannot be lower than one because the net interest rate would be negative. In addition, $R$ cannot be greater than two because if so, the net interest rate would be greater than 100%, which is not common in the data. Then, $R$ has a lower and upper bound. In particular, I assume that $R$ in $[1,1.08]$ with a Markov transition probability function associated to (5) as $q(R', R)$ . Importantly, innovations in interest rates are crucial for firms since most bank loans have floating rates tied to monetary policy rates. This connection allows monetary policy to affect the liquidity and investment decisions of firms (e.g. Ippolito et al. Reference Ippolito, Ozdagli and Perez-Orive2018).

An important characteristic of working capital loans is that they are a kind of short-term debt. The firm needs cash to finance the labor payments at the beginning of the period before the realization of profits which are obtained at the end of the period. Then, the working capital loans are paid with the cash flows derived from revenue. As a result, working capital loans can be considered short-term or intra-period debt. Additionally, it is common to observe in the data that banks do not require collateral for working capital loans. Because of that, I do not consider a collateral constraint for working capital loans in the model as previous studies do (Mahmoudzadeh et al. Reference Mahmoudzadeh, Nili and Nili2018).

To illustrate the relevant role of the working capital channel in the firm’s behavior, I describe what happens in the economy when there is a positive interest rate shock. The first effect of this shock is the increase in the working capital cost. The firm faces a tradeoff. On the one hand, the firm could reduce its working capital demand by reducing labor demand. This reduces the total variable cost and hence increases the cash flow. On the other hand, with lower labor, the production decreases, negatively affecting the investment and the next period’s capital stock. With more insufficient capital stock in the next period, the profits would be lower, negatively affecting the expected discount value of cash flow. Furthermore, if the shock persists over time, it will contribute dynamically to pushing down labor in the future. This strengthens the initial effects on discounted expected cash flow. Therefore, the manager must decide the optimal value of the labor force, balancing the benefits and costs in the presence of gross interest rate shock.

Figure 2. Working capital channel. Positive interest rate shock.

The proportion of $wn$ financed by working capital loans, $\phi$ , is essential to determine the relevance of the interest rate shock on a firm’s decisions. Figure 2 shows the dynamic effects of the interest rate shock through a working capital channel. This channel is controlled by $\phi$ . Two extreme cases emerge in this model. The first is when a firm does not need money in advance to finance the variable cost. In this case, $\phi$ is zero, and the working capital channel is irrelevant to transmitting effects from interest rate shocks. In fact, in this setting, the interest rate does not affect the firm’s decisions. The second case is when a firm finances all its variable costs with working capital loans. In this case, $\phi$ is one, and the firm is sensitive to any movement in the interest rate. Therefore, any movement in the interest rate affects a firm’s decisions, so the working capital channel is very important. In this context, the relevant empirical questions are as follows: How big is $\phi$ for the entire economy? And is $\phi$ different across industries? Given the relevance of the working capital channel, measuring $\phi$ is of the first order of importance.

3.3 The firm’s objective function

Because I assume there are no agency costs between shareholders and managers; there is no difference between the manager’s and shareholders’ objective functions. In particular, the risk-neutral manager maximizes the cash flows, $d$ , that go to shareholders. Under the standard accounting identity, I can express distributions to shareholders as

(6) \begin{equation} d(k,k',n,R) \equiv zk^{\alpha }n^{1-\alpha } - C(I) - \text{cash} - R_{-1}\text{loan}_{-1} \end{equation}

The first term of equation (6) represents operating profits at the end of the period $t$ . These profits are then spent on investment $C(I)$ , on cash, and on paying the previous debt, $R_{-1}\text{loan}_{-1}$ . That debt was obtained at beginning Footnote ⁶ of the period $t$ since the firm needs to finance its variable cost in advance (working capital requirements).

The common assumption in monetary policy models is that the entire labor cost $ w'n'$ is financed in advance only with loans: $loan= w'n'$ . I relax this strong assumption by assuming that working capital loans finance a fraction $\phi$ of the total labor cost.

(7) \begin{equation} \text{loan} = \phi w'n' \end{equation}

My goal is to allow the data to tell us what the value of $\phi$ is across industries and the entire Compustat sample. Additionally, complementary to loan financing, I allow the firm to finance its working capital requirement with cash. As a result, the firm uses cash and loans at the current period to pay its future labor costs in advance.

(8) \begin{equation} \text{cash} + \text{loan} \geq w'n' \end{equation}

Equation (8) reflects that the firm faces a cash-in-advance constraint since there exists a mismatch between the working capital payment at the beginning of the period and the realization of the profits at the end of the same period. Considering equation (7) for $t$ and $t-1$ and equation (8) with equality into the shareholder’s distribution identity (equation 6), we have:

(9) \begin{equation} d(k,k',n,n',R) \equiv zk^{\alpha }n^{1-\alpha } - C(I) - R_{-1}\phi w n - (1-\phi )w'n', \end{equation}

where $R_{-1}\phi w n$ is the interest and principal payment of the working capital loans in $t-1$ that should be paid in $t$ . Additionally, $(1-\phi ) w'n'$ represents the fraction of the working capital requirement financed with cash in $t$ . I assume that investment can be financed by internal sources, after paying working capital loans, and by issuing equity (negative $d$ ).

Furthermore, if there is equity issuance, the shareholders incur an issuing cost $\lambda$ . A positive firm’s cash flow is distributed to its shareholders, while a negative cash flow implies that the firm obtains funds from shareholders. In the latter case, the firm would pay a linear cost, $\lambda$ . Thus, the shareholder’s final cash flows are given by

(10) \begin{eqnarray} d^*(k,k',n,n',R) &=& d(k,k',n,R), \quad \quad \text{if} \quad d(k,k',n,R) \geq 0 \end{eqnarray}

(11) \begin{eqnarray} d^*(k,k',n,n',R) &=& (1+\lambda )d(k,k',n,R), \quad \quad \text{if} \quad d(k,k',n,R)\lt 0 \end{eqnarray}

In a compact form, $d^*(k,k',n,n',R)$ would be:

(12) \begin{equation} d^*(k,k',n,n',R) = d(k,k',n,n',R)\cdot (1+\lambda \cdot 1_{d\lt 0}) \end{equation}

The manager’s objective function is the expected present value of cash flows given by (12), which can be expressed recursively as

(13) \begin{equation} V(k,n,R) = \underset{\{k',n'\}}{\text{max}}\left\{d^*(k,k',n,n',R) + E[S'\cdot V(k',n',R')] \right\} \end{equation}

in which $V(k,n,R)$ is the value of the firm’s equity and $S' = \beta c_{t}/c_{t+1}$ is the stochastic discount factor,Footnote ⁷ which the manager uses to discount the firm’s cash flows. The manager makes choices $\{k',n' \}$ so that the shareholders obtain the maximum cash flows. Assuming a finite state space $A$ , $E[S'\cdot V(k',n',R')]$ can be expressed as $\sum _{A} [q(R',R) \cdot S' \cdot V(k',n',R')]$ in which $q(R',R)$ represents the probability of jumping from one state in $t$ to another state in $t+1$ .

3.4 Firm’s policy functions

The main goal of this paper is to understand how the working capital channel affects the firm’s decisions by estimating this structural model directly. With this goal in mind, two steps are important in analyzing the economic implications of the model carefully. The first is to understand the estimation results and the second is to identify the model parameters. Both steps require understanding the economics behind the model. To do this, I analyze the manager’s maximization problem by examining the first-order conditions for optimal investment and labor. From first-order conditions, I obtain the labor demand expressed as

(14) \begin{equation} (1+\lambda 1_{d\lt 0})(1-\phi )w' = E\left[ S'\left( (1-\alpha )z'k'^{\alpha }n'^{-\alpha } - R \phi w' \right)\left(1+\lambda 1_{d'\lt 0}\right)\right] \end{equation}

To explain the intuition of the role of the interest rate on labor demand, I consider equation (14) without the equity issuance friction ( $\lambda =0$ ). As a result, the labor demand becomes:

(15) \begin{equation} \underbrace{(1-\phi )w' + E\left[ S' R \phi w'\right]}_{\substack{\text{Present value of the} \textbf{marginal cost} \\ \text{of 1 additional unit of labor}}} = \underbrace{E\left[ S'\left( (1-\alpha )z'{k'}^{\alpha }{n'}^{-\alpha } \right)\right]}_{\substack{\text{Present value of the}\, \textbf{marginal benefit} \\ \text{of 1 additional unit of labor}}} \end{equation}

The left side of equation (15) is the present value of the marginal cost of an additional labor unit. Two terms compound it: the first is the fraction of the cost of one labor unit financed with cash $(1-\phi )w'$ , and the second is the present value of the loan that the firm should pay to finance the remaining fraction of labor cost $E\big [ S' R \phi w'\big ]$ . The firm compares that marginal cost with the present value of the marginal benefit of an additional labor unit (right side of equation 15). This marginal benefit is the marginal productivity of labor represented by $(1-\alpha )z'{k'}^{\alpha }{n'}^{-\alpha }$ .

Equation (15) shows the direct effect of the working capital channel on labor demand. An unexpected increase in interest rate pushes up the marginal cost of labor and hence decreases the labor demand. The effect of that shock is controlled by $\phi$ —the proportion of variable cost $ w'n'$ financed by working capital loans. The second critical equation is the optimal investment which is expressed as

(16) \begin{equation} \left[1_{I\geq 0} + \theta 1_{I\lt 0}\right]\left(1 + \lambda 1_{d\lt 0}\right) = E\left[S'\left[\alpha z'k'^{\alpha -1}n'^{1-\alpha } + (1-\delta )\left(1_{I'\geq 0} + \theta 1_{I'\lt 0}\right)\right]\left(1 + \lambda 1_{d'\lt 0}\right)\right] \end{equation}

If the investment is totally reversible ( $\theta =1$ ) and there is no friction in issuing equity ( $\lambda =0$ ), the equation (16) turns out a standard investment equation without frictions:

(17) \begin{equation} 1 = E\left[S'\left(\alpha z'k'^{\alpha -1}n'^{1-\alpha } + (1-\delta )\right)\right] \end{equation}

Two additional equations. Furthermore, it is necessary to define two additional conditions to close the model. The first one is the equilibrium in the goods market, which is represented by

(18) \begin{equation} y = c + I, \end{equation}

in which $y$ is the firm’s production function $zk^{\alpha }n^{1-\alpha }$ or revenues, $c$ is consumption, and $I$ is the investment as defined in the previous subsection. The labor supply is the second condition to define the equilibrium in the labor market. I assume that the labor supply is characterized as

(19) \begin{equation} w = \frac{\theta _n c}{1-n}, \end{equation}

in which $w$ is the real wage, $\theta _n$ is a parameter that measures the relevance of labor (or leisure) in a utility function, and $n$ is labor. I summarize the model’s equations in Table A1, shown in Appendix A.2.

4.1 Numerical policy functions

In this section, I examine the policy functions $\{k',n' \} = g(k,n,R)$ to gain more insight about the model. Furthermore, the firm value $v$ —which is calculated as the expected discount cash flows $v = \sum _{t=0}^{\infty } S_t d_t$ —is analyzed as well as profitability $y/k$ , investment rate $i/k$ , and working capital ratio $\phi w'n'/k$ . The firm’s optimal response expressed in the dynamic behavior of these variables allows us to understand how the working capital channel transmits the interest rate shock to the firm’s decisions.

To analyze the policy functions, it is necessary to assign the corresponding value to every model parameter. The literature related to dynamic models suggests two main approaches to do so. The first is to estimate these parameters, and the second is to calibrate them. Since I am interested in estimating the parameter that controls the working capital channel—that is, $\phi$ , I am using the structural model and data from Compustat for estimating it, which is explained carefully in the following sections. However, since this section aims to understand how interest shock affects firm decisions through the working capital channel, I calibrate all parameters according to previous studies and assume an intermediate value of $\phi$ ( $\phi =0.5$ ). This last assumption means the firm finances half its variable cost with working capital loans.

4.1.1 Calibration

The average share of capital in total production, $\alpha$ , is around 0.77 according to Nikolov and Whited (Reference Nikolov and Whited2014). The value of the depreciation rate is 10% annually, a standard assumption in business cycle literature. The discount factor parameter $\beta$ is 0.96 according to the steady-state value of gross interest rate $R_{ss} = 1.04$ . The equity issue cost as the percent of distributions $\lambda$ is 0.04 based on Michaels et al. (Reference Michaels, Page and Whited2019), and I also use their estimate for the parameter that drives investment irreversibility $\theta$ , which is 0.534. Christiano et al. (Reference Christiano, Trabandt and Walentin2010) suggest that the persistence and the standard deviation of the interest rate shock in terms of monetary policy are 0.87 and 0.51, respectively. However, not all the volatility of the monetary shock is transmitted to the interest rate of loans. As a result, I assume that the relevant volatility of the interest rate shock for the firm is one-third of the corresponding monetary policy, but the persistence is the same. In other words, I assume that $\rho$ and $\sigma _{\varepsilon }$ are 0.87 and 0.51/3, respectively. Additionally, I calibrate $\theta$ —which measures the relevance of labor (or leisure) in the utility function—to be consistent with the value of the steady state of labor $n_{ss} = 0.2$ . As a result, $\theta _n$ is equal to 3.36. Finally, I do not consider income effects on labor supply. To do that, I keep the level of consumption at its steady-state value ( $C_{ss} = 0.25$ ), which allows us to get the consumption-output ratio in steady state equal to 75% and the investment-output ratio in steady state equal to 25%. Considering that consumption takes its steady-state value all the time, the upward-sloping labor supply curve under interest rate shocks allows us to study the effects of this shock on labor through the response of labor demand.

Figure 3. Policy function. The figure depicts the optimal response of the labor, investment rate, profitability, and working capital ratio in response to the interest rate shock $R$ for every level of capital. Low R, medium R, and high R correspond to $R=1$ , $R=1.04$ , and $R=1.08$ respectively.

4.2 Analysis of working capital channel

A crucial quantitative question is how big the working capital channel is. I address this question with annual data for firms listed in Compustat, which will be discussed later. Before that, it is important to understand how the endogenous variables respond in different settings of the working capital channel. Figure 4 shows three cases for the working capital channel. In the first one, this mechanism is absent. Hence any movement in interest rate does not affect the firm’s decisions, and all variables remain in their steady-state values. This setting corresponds to the case in which the firm finances all its variable costs with cash, which is paid at the end of the period. In this case, we do not have the cash flow mismatch problem.

Figure 4. Relevance of working capital channel in the firm’s decisions (Impulse response function). This figure shows how a firm’s value, investment rate, profitability, and working capital ratio behave in three cases: no working capital channel ( $\phi =0$ ), a moderate working capital channel ( $\phi =0.5$ ), and a full working capital channel ( $\phi =1$ ).

The second case is when the firm finances in advance, at the beginning of the period, 50% of its variable costs with working capital loans ( $\phi =0.5$ ). In this case, the interest rate shock affects the economy through the working capital channel. As shown in the bottom-left-side panel of Figure 4, the firm optimally reduces its labor demand since the labor marginal cost has increased. The underlying effect is the reduction of revenue and, hence, profits. All of these effects reduce the current and future cash flows with a negative impact on the firm’s value, as shown in Figure 4.

The last case is when a firm finances all its variable costs in advance with working capital loans. In this case, the effect of interest rate shock is stronger than in previous cases since the working capital channel transmits all the shock to the firm’s decisions. As a result, the labor demand, firm’s value, profitability, and investment rate decrease significantly (see Figure 4).

5.1 Data

The data come from Compustat. I consider a sample period from 1971 until 2018 with annual frequency. The sample does not consider firms associated with financial services, utilities, and government administration. Furthermore, I eliminated any row in the sample where the main variables (operating income, total assets, capital expenditure, and debt in current liabilities) have no information. Additionally, I winsorize profitability, investment rate, and the working capital ratio at the 1st and 99th percentiles. After these filters, the final panel has 86,911 observations for 5,739 firms from 1971 to 2018 at an annual frequency. I consider working capital loans as Debt in Current Liabilities. This variable represents liabilities due within one year, including the current portion of long-term debt. In particular, this variable is the sum of accounts payable, other current liabilities, debt in current liabilities, and income taxes. Table 2 contains the variable definitions.

Since there exists the possibility that the working capital ratio varies across industries, I split the sample by type of industry. Table 2 shows descriptive statistics for the entire sample and every industry.

Table 2. Data definitions

5.2 Identification

Model identification is a cornerstone of the SMM technique. The identification requires choosing moments sensitive to variations in the structural parameters. In technical terms, identification requires that the relationship between the model parameters and the moments be one-to-one and onto. I illustrate the identification procedure by an example in Appendix B.2.

I now describe six potentially informative moments to identify the three parameters of interest ( $\phi$ , $\alpha$ , and $\delta$ ). In particular, I explore two moments (mean and variance) associated with three main variables: profitability, investment rate, and working capital ratio. I chose these variables due to their connections with the structural parameters in the theoretical model and because I have available data in Compustat, which allows me to construct these variables.

5.2.3 Identification of $\delta$

In the case of $\delta$ , which represents the capital depreciation rate, the first moment of investment rate, profitability, and working capital ratio are informative to identify this parameter. In particular, the mean of the investment rate seems to be more informative due to varying significantly over the range of $\delta$ . Furthermore, as shown in Figure B5 (see Appendix B.2), the variances of both profitability and working capital ratio are monotonic but with minor changes. Consequently, they are not informative. Finally, the variance of investment rate does not vary, avoiding the identification of $\delta$ . The identification analysis of $\delta$ suggests that the mean of profitability, investment rate, and working capital contain information to identify this parameter.

Table 3. Structural parameter estimates

This table presents the estimated structural parameters, with standard errors in parenthesis; $\alpha$ is the capital share in total profits, $\delta$ is the depreciation rate, and $\phi$ is the proportion of working capital which is financed by working capital loans. I estimate these parameters for the entire sample and seven industries (Agriculture, Mining, Construction, Manufacturing, Wholesale Trade, Retail Trade, and Services)

*This industry includes forestry and fishing as well.

I summarize the identification process in Table 3. One conclusion is that the mean of the working capital ratio is connected to three parameters: $\alpha$ , $\phi$ , and $\delta$ . Additionally, the profitability mean depends on $\alpha$ and $\delta$ . Furthermore, $\delta$ is related to the mean of the investment ratio. I express these relationships between moments and parameters in the following equations to see how the identification process works.

(20) \begin{eqnarray} \text{Mean(working capital ratio)} &=& f_1(\alpha, \phi, \delta, \Psi ) \end{eqnarray}

(21) \begin{eqnarray} \text{Mean(profitability)} &=& f_2(\alpha, \delta, \Psi ) \end{eqnarray}

(22) \begin{eqnarray} \text{Mean(investment rate)} &=& f_3(\delta, \Psi ), \end{eqnarray}

where $f_1$ , $f_2$ , and $f_3$ represent functions that map moments to parameters. Potentially, moments could depend on several parameters, not only those I want to study, such as $\alpha$ , $\phi$ , and $\delta$ . Then, the variable $\Psi$ contains all the remaining parameters present in the model, which are calibrated, such as the persistence of interest rate $\rho$ and the standard deviation of the innovation of interest rate $\sigma _{\varepsilon }$ . As I mentioned in the paragraphs above, the identification process requires that moments of some variables are related to the parameters that I want to estimate. In this case, we can proceed recursively. From equation (22), we can obtain $\delta$ . Next, from equation (21), we can exactly identify $\alpha$ since we know $\delta$ . Equation (20) allows me to identify the value of $\phi$ given the value of $\delta$ and $\alpha$ . In conclusion, for identifying the three parameters ( $\alpha$ , $\phi$ , and $\delta$ ), I can use three informative moments: the mean of working capital ratio, the mean of profitability, and the mean of investment ratio. The model is exactly identified since I have three parameters and three moments. Given that the variance of profitability reacts to all three parameters to different degrees, I will use it in the estimation process. The model is overidentified in that last case since I have more moments than parameters.

5.3 Estimation

I estimate the model parameter of the working capital channel $\phi$ , the capital-output elasticity $\alpha$ , and the depreciation rate $\delta$ using SMM. The remaining parameters of the model are calibrated based on previous studies. The SMM estimation technique is well-known in econometric literature. Its basic idea is to adjust the parameter of interest (for instance, $\beta$ ) to get similar properties for the observed endogenous variables, $y_t$ , and their simulated counterparts, $y_t^s$ (Gouriéroux and Monfort, Reference Gouriéroux and Monfort1996). In particular, SMM finds $\theta$ such that the empirical moments of variables $y_t$ are as close as possible to the moments of the simulated variable $y_t^s$ , which come from the structural model. To be explicit about how I am applying this procedure to the structural model described before, I split the procedure into two sets of steps in the spirit of Strebulaev and Whited (Reference Strebulaev and Whited2012): moments and estimation procedure (see Figure 7 in Appendix B).

First, I choose a set of moments that I initially want to match. Since in the model, I have three main variables (profitability, investment rate, and working capital ratio), the moments chosen are the average and variance of these variables. As a result, I have six moments. In this step, I do not know if I will finally require the model to match all six moments. The identification process will tell us what moments of the six we need.

Second, I identify what moments are relevant to estimate the three parameters $\phi$ , $\alpha$ , and $\delta$ . From the six moments chosen in the first step, I evaluate which provides information about the parameters. Since I am estimating three parameters, I must choose at least three moments from the available six.

The third step is to simulate the chosen moments from the identification step and save them in $m(y^s,\beta )$ and do the same for variables in the real data and save their moments in $M(y)$ . To obtain the simulated moments, I choose the starting value of the set of parameters. The next step is to calculate the covariance matrix of the empirical moments. The inverse of this matrix represents the GMM weight matrix. I denote this matrix as W.

Thus far, we have a subset of moments related to the parameters of interest $\phi$ , $\alpha$ , and $\delta$ . Furthermore, we have empirical and simulated moments and the GMM weight matrix. Now, I begin the estimation process. Specifically, SMM chooses $\phi$ , $\alpha$ , and $\delta$ such that these parameters minimize the SMM objective function $Q(y,y^s,\phi,\alpha,\delta )$ . This function is the sum of the square of the difference between the empirical moments and the simulated moments weighted by the inverse of the covariance matrix of empirical moments $W$ . $Q({\cdot})$ is defined as follows.

(23) \begin{equation} Q(y,y^s,\phi,\alpha,\delta ) \equiv (M(y) - m(y^s,\phi,\alpha,\delta ))' W (M(y) - m(y^s,\phi,\alpha,\delta )) \end{equation}

After estimating $\phi$ , $\alpha$ , and $\delta$ , I adjust the standard errors and test statistics for simulation error. Finally, I use a specification test, which refers to a general test of the overidentifying restrictions of the model, which can be written as:

\begin{equation*}\frac {NJ}{1+J}Q(y,y^s,\phi,\alpha,\delta )\end{equation*}

In which $J$ is the ratio of the number of observations in the simulated data to the number of observations in the real data $N$ .