Silviase/jgov_v1.4_full_ann huggingface dataset:
$ K_{t-1} $ : real capital stock at the end of period t-1

$ z_{t} $ : expected present value of current and future investment allowances on a unit of investment expenditure in period t

The fundamental part of the stock price is proxied by the present discounted value of marginal profitability of investment divided by the investment goods deflator. This is tax-adjusted marginal q and it is defined as:

$$ (4)MAQ_{t}=\frac{1}{(1-z_{t})p_{t}^{I}}E_{t}\Bigg[\sum_{j=0}^{\infty}\beta_{t+j}(1-\delta)^{j}(1-\tau_{t+j})\pi_{t+j}\Bigg] $$

$$ where\quad\beta_{t+j}\equiv\prod_{i=1}^{j}(1+r_{t+i})^{-1}\quad(j=1,2,\cdots)\quad,\beta_{t}\equiv1 $$

$ r_{t+1} $ : one-period nominal discount rate in period t

$ \tau_{t} $ : corporate tax rate

$ \pi_{t} $ : before-tax profits divided by the real capital stock

$E_{t}[\bullet]$: mathematical expectation operator conditional upon the information available in period t

In deriving eq.(4) we assume the technology of constant returns to scale. Then the use of average profit rates as the marginal profits can be justified. The marginal q is not directly observable since it includes the unobservables: future stream of discount rates and profit rates. Therefore to make eq.(4) operational we have to know the stochastic structure underlying the discount factor and profit rates. $ ^{12)} $ As a first step to pinpoint the stochastic process we test whether the discount factor $ d_{t}\left(=(1-\delta)/(1+r_{t})\right) $ and profit rates $ \pi_{t} $ have unit roots. Two types of unit root tests are conducted. One is the Augmented Dickey Fuller test(ADF test) and the other is Phillips-Perron test. $ ^{13)} $ The null hypothesis is chosen as one unit root without trend for the discount factor and one unit root with trend for the profit rates. Technical progress may have some effect on the level of profit rates, which will justify our allowance for trend in the case of profit rates. The lag length is chosen to be 4 in both types of tests. The results of unit root tests are presented in Table 4. As for the discount factors, we cannot reject the null hypothesis for any industries, while the unit root test is decisively rejected at the 5 % significance level for all the industries by the Phillips-Perron test. It implies that the movement of profit rates are better characterized by trend stationary process.

Based on the test results, we specify the stochastic process of the discount factors and the profit rates as follows: $ ^{14)} $
$$ \Delta d_{t}=m_{10}+\sum_{k=1}^{p}m_{1k}\Delta d_{t-k}+\mathfrak{E}_{1t} $$

$$ (6)\boldsymbol{\pi}_{t}=m_{20}+\sum_{k=1}^{q}m_{2k}\boldsymbol{\pi}_{t-k}+n_{2}\boldsymbol{t}+\boldsymbol{\varepsilon}_{2t} $$

$$ \begin{aligned}where t&:time trend\\\varepsilon_{1t},\varepsilon_{2t}&:white noise\end{aligned} $$

The lag lengths of eqs. $ (5) $ and $ (6) $ are determined on the basis of Schwarz Information Criterion, which are shown in Table 5. The lag length of one is chosen for the discount factors in all the industries but construction and machinery. On the other hand, the lag length is equal or longer for the profit rates than the discount factors. Table 6 shows the estimation results of eqs. $ (5) $ and $ (6) $ based on the lag length chosen above.

When the stochastic process of $ d_{t} $ and $ \pi_{t} $ is characterized by eqs.(5) and (6), it can be shown that the tax-adjusted marginal q is written as: $ ^{15)} $

$$ \begin{aligned}(7)MAQ_{t}=&\left[\frac{\pi_{t-1}}{\left(1-d_{t-1}\right)^{2}}\mathbf{c}^{\prime}(\mathbf{I}-d_{t-1}\mathbf{M_{1}})^{-1}\mathbf{M_{1}}\mathbf{a}_{t-1}+\frac{\pi_{t-1}}{\left(1-d_{t-1}\right)^{3}}\mathbf{c}^{\prime}(\mathbf{I}-d_{t-1}\mathbf{M_{1}})^{-1}\boldsymbol{\theta}_{1}\right.\\&\left.+\mathbf{c}^{\prime}(\mathbf{I}-d_{t-1}\mathbf{M_{2}})^{-1}\mathbf{M_{2}}\mathbf{b}_{t-1}+\frac{1}{\left(1-d_{t-1}\right)}\mathbf{c}^{\prime}(\mathbf{I}-d_{t-1}\mathbf{M_{2}})^{-1}\boldsymbol{\theta}_{2}\right.\\&\left.+\left\{\frac{d_{t-1}}{\left(1-d_{t-1}\right)^{2}}+\frac{t}{\left(1-d_{t-1}\right)}\right\}\mathbf{c}^{\prime}(\mathbf{I}-d_{t-1}\mathbf{M_{2}})^{-1}\mathbf{n}\right]\frac{\left(1-\boldsymbol{\tau}_{t}\right)}{\left(1-z_{t}\right)p_{t}^{I}}\end{aligned} $$

$$ \begin{aligned}where\quad\mathbf{c}^{\prime}&=(1,0,\cdots,0)\\\mathbf{a}_{t-1}&=(\Delta d_{t},\Delta d_{t-1},\cdots,\Delta d_{t-p+1})\\\boldsymbol{\theta}_{1}&=(m_{10},0,\cdots,0)\\\mathbf{b}_{t-1}&=(\boldsymbol{\pi}_{t},\boldsymbol{\pi}_{t-1},\cdots,0)\\\boldsymbol{\theta}_{2}&=(m_{20},0,\cdots,0)\\\mathbf{n}^{\prime}&=(n_{2},0,\cdots,0)\\\mathbf{M}_{1}&=\begin{bmatrix}{{{m_{11}}}}&{{{m_{12}}}}&{{{\cdots}}}&{{{m_{1,p-1}}}}&{{{m_{1p}}}} \\{{{1}}}&{{{0}}}&{{{\cdots}}}&{{{0}}}&{{{0}}} \\{{{0}}}&{{{1}}}&{{{\cdots}}}&{{{0}}}&{{{0}}} \\{{{\vdots}}}&{{{\vdots}}}&{{{\ddots}}}&{{{\vdots}}}&{{{\vdots}}} \\{{{0}}}&{{{0}}}&{{{\cdots}}}&{{{1}}}&{{{0}}}\end{bmatrix}\end{aligned} $$
$$ \mathbf{M}_{2}=\left[\begin{aligned}m_{21}&\quad m_{22}&\cdots&\quad m_{2,q-1}&\quad m_{2q}\\ 1&\quad0&\cdots&\quad0&\quad0\\ 0&\quad1&\cdots&\quad0&\quad0\\ \vdots&\quad\vdots&\ddots&\quad\vdots&\quad\vdots\\ 0&\quad0&\cdots&\quad1&\quad0\end{aligned}\right] $$

We use the ratio of operating profits to the real capital stock as a proxy of $ \pi $ . Before examining the relationship between the market valuation and the fundamentals, we give some summary statistics on the AVQ and the MAQ variables. The mean and standard deviation are computed for the period of the first quarter of 1970 to the fourth quarter of 1990. They are shown in Table 7. The magnitude of the AVQ variable is around unity for all industries(1.2135), construction(1.8020), manufacturing(1.5899), food and beverages(1.3981), and textiles(0.9151), while that of the MAQ variable hovers around unity for all industries(1.2493), manufacturing(1.1631), food and beverages(1.3952), textiles(1.1069), and real estate(1.5040). The average q is negative for pulp, paper and paper products(-0.0849) and real estate(-0.1896). Both the AVQ and the MAQ variables are well above unity for chemistry (2.5548, 2.0450), steel (3.3800, 2.0541), machinery(2.7670, 3.6088), electrical machinery (7.6279, 4.3456), and wholesale and retail trade (5.1149, 9.8541).

It should be noted that the AVQ variables are more volatile than the MAQ variables across all the industries but pulp, paper, and paper products in the sample. In fact the standard deviation of the former is 1.71 to 5.56 times as large as the latter(third column of Table 7).

Now we examine the relationship of the fundamentals with the market valuations. If the stock price reflects all the relevant information on the future profitability of investment projects undertaken by the firm, then the AVQ variable will be equal to the MAQ variable. Once both series of the AVQ and the MAQ variables are constructed, it is easy to test equality between two series. Taking the sampling errors into consideration, the null hypothesis is written as follows:

(8) $ AVQ_{t} = MAQ_{t} + u_{t} $

where $ u_{t} $ : sampling error uncorrelated with the MAQ variable

Eq.(8) can be tested in several ways. $ ^{[16]} $ The simplest way to test eq.(8) is to estimate the following equation and test the hypothesis that $ \alpha_{0}=0 $ and $ \alpha_{1}=1 $ .

(9) $ AVQ_{t} = \alpha_{0} + \alpha_{1}MAQ_{t} + u_{t} $

The ordinary least squares (OLS) estimation was conducted for each industry. The regression results
of eq.(9) are given in Table 8 and the test statistics are presented in Table 9. The null hypothesis is rejected at the 1% significance level for all the industries except all industries, food and beverages, and textiles. Although the null hypothesis is not rejected statistically for all industries, food and beverages, or textiles, it seems to us that the results are as a whole unfavorable to the view that the market valuations reflect the fundamentals for the following reasons.

First the statistical fitness is quite poor. The adjusted coefficient of determination is at most 0.6980 for construction and it is nearly zero or negative for most of the industries.

Second the Durbin-Watson statistics for all the cases are quite low, indicating the possibility of positive serial correlation. This may hint the possibility that the stock price is driven by other factors besides the fundamentals. $ ^{[17]} $ Drawing the difference between the AVQ and the MAQ variables $ (AVQ_{t}-MAQ_{t}) $ is useful for searching for the candidates to cause this serial correlation. Our strategy is to see whether there still remains systematic parts in the market valuation even after taking the stock market efficiency into consideration. The difference is depicted in Figure 10-1 to 10-14. There exists a pattern of persistent positive residuals in the middle of 80's for all industries, manufacturing, food and beverages, textiles, chemicals, steel, non-ferrous metal, machinery, and electrical machinery. The divergence of the market valuations from the fundamentals is notably large during the third quarter of 1986 to the first quarter of 1990 for most of the industries. This pattern indicates an overvaluation in the stock market during these periods, which may be identified as the evidence for the bubbles or fads in the stock price.

Then the next question to be posed is: Is this persistent divergence of the stock price from the fundamentals related to any economic factors? To answer this question we estimate the following equation:

$$ (10)AVQ2_{t}=\alpha_{0}+\alpha_{1}MAQ_{t}+\alpha_{2}\frac{p_{t}^{L}L_{t-1}}{(1-\delta)(1-z_{t})p_{t}^{I}K_{t-1}}+u_{t} $$

$$ where\quad AVQ2_{t}=\frac{V_{t}+B_{t-1}-A_{t}}{(1-\delta)(1-z_{t})p_{t}^{I}K_{t-1}} $$

Equation (10) is obtained simply by moving the term including the market value of land from the LHS in eq.(9) to the RHS. When the stock price reflects correctly the market value of land held by the firm as well as the prospects of investment projects, then not only $ \alpha_{1} $ but also $ \alpha_{2} $ will be unity.

Suppose, however, that $ u_{t} $ is positively correlated with the land value held by the firm. Then application of the OLS to eq.(10) will yield biased estimates. Especially, the parameter estimate of $ \alpha_{2} $ will be upward biased. Therefore estimates of $ \alpha_{2} $ in excess of unity may suggest that the nonfundamentals are positively correlated with the market value of land held by the firm. The estimation
results of eq.(10) is given by Table 10. It is clear that the parameter estimates of $ \alpha_{2} $ is upward biased for all the industries except real estate, supporting the positive association of the bubbles or fads with the land value of the firm.

Now that the divergence of the market valuation from the fundamentals is clear, next task is to investigate the response of investment to the fundamentals and the non-fundamentals. This is the topic to be pursued in the next section.

### 5. Portfolio Behavior of the Corporation and Business Fluctuations

In the preceding sections we shed light on two aspects of the asset markets which have important implications to the fluctuations of business cycle. One is the role of collateralizable assets of borrowers in loan contract, which is expected to affect the level of investment. The other is the divergence of the stock price from the fundamentals and our conjecture is that the fundamentals and the non-fundamental parts might have different effects on the real economy.

We examine these points empirically. Specifically we confine our attention to the portfolio behavior of the corporations and analyze how the portfolio is affected by an increase of collateral value of assets and quantify the effects of the fundamentals and the non-fundamentals parts in the stock price on the portfolio allocation. The data set is the same as employed in section 3. The sample period used for estimation ranges from the second quarter of 1970 to the fourth quarter of 1990. We use OLS as an estimation method. Based on the industry data, we estimate the portfolio equations of the corporations for three assets: physical investment, land purchase, and borrowings.

To start with, we derive the investment function in the benchmark case based on the neoclassical intertemporal model of the firm. The firm chooses the optimal level of investment as well as the current inputs and production so that the value of firm can be maximized. The value of firm is defined as the discounted sum of future dividends $ (D_{t}) $ . In other words,

$$ V_{t}=E_{t}\Biggl[\sum_{j=0}^{\infty}\beta_{t+j}D_{t+j}\Biggr] $$

$$ \begin{aligned}where\quad\beta_{t+j}=&\prod_{i=1}^{J}(1+r_{t+i})^{-1}\quad(j=1,2,\ldots)\quad,\beta_{t}\equiv1\\D_{t}=&(1-\tau_{t})\Big[p_{t}\Big\{F\left(K_{t},N_{t}\right)-G(\mathbf{I}_{t},K_{t})\Big\}-w_{t}N_{t}-i_{t}B_{t-1}+i_{t}^{L}L_{t-1}\Big]\\&+NB_{t}-(1-z_{t})p_{t}^{I}I_{t}-p_{t}^{L}IL_{t}+\widetilde{A}_{t}\end{aligned} $$

where $ r_{t+1} $ : one-period discount rate in period t

$ p_{t} $ : output price

$ N_{t} $ : labor input
$ I_{t} $ : real investment in plant and equipment

$ K_{t} $ : capital stock at the end of period t

$ p_{t}^{I} $ : investment goods price in period t

$ B_{t-1} $ : net financial liability at the end of period t-1

$ NB_{t} $ : increase of net financial liability in period t

$ w_{t} $ : wage rate

$ z_{t} $ : present value of investment allowances on a unit of investment expenditure in period t

$ i_{t}^{L} $ : rent on land

$ i_{t} $ : interest rate on $ B_{t-1} $

$ \widetilde{A}_{t} $ : value of the depreciation allowances on investment made before period t

that can be claimed in period t

$ IL_{t} $ : real purchase of land in period t

$ \tau_{t} $ : corporate tax rate in period t

The production technology of the firm is represented by the production function of $ F \left( K_{t}, N_{t} \right) $ . We assume that the firm faces convex adjustment costs, $ G(I_{t}, K_{t}) $ , in changing its capital stock. The relationship between the stock and the flow of assets is written as

(12) $ K_{t} = (1 - \delta)K_{t-1} + I_{t} $

(13) $ L_{t}=L_{t-1}+IL_{t} $

(14) $ B_{t} = B_{t-1} + NB_{t} $

Maximization of eq. $ (11) $ subject to eqs. $ (12) $ , $ (13) $ , and $ (14) $ yields the following investment function: $ ^{[18]} $

$$ \left(\frac{I_{t}}{K_{t-1}}\right)=\alpha_{0}+\alpha_{1}(MAQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\tau_{t})p_{t}} $$

Furthermore, when both the production technology and the adjustment costs function have the property of linear homogeneity, the fundamentals are equal to the market valuations, so that eq.(15) is simplified as:
$$ (16)\left(\frac{I_{t}}{K_{t-1}}\right)=\alpha_{0}+\alpha_{1}(AVQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\tau_{t})p_{t}} $$

Eq. $ (16) $ is the investment demand equation frequently estimated in the literature. Investment-capital ratio is an increasing function of the tax-adjusted average q. We call this model as Model 1.

This model is extended in two directions. Firstly when the firm faces the liquidity or borrowing constraints, the level of investment is determined by the level of cash flow. Secondly in the formulation above land is treated as an asset yielding rents, not as a production factor. However, as is seen above, it is likely that land plays a vital role as a collateral to easy the credit conditions. If it is the case, then it is expected that the market value of land held by the firm will have a positive effect on investment. $ ^{19)} $

Taking account of these modifications, Model 1 is rewritten as

$$ (17)\left(\frac{I_{t}}{K_{t-1}}\right)=\alpha_{0}+\alpha_{1}(AVQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\mathfrak{r}_{t})p_{t}}+\alpha_{2}\frac{p_{t}^{L}L_{t-1}}{p_{t}^{I}K_{t-1}}+\alpha_{3}\frac{CF_{t}}{p_{t}^{I}K_{t-1}} $$

where $ CF_{t} $ : cash flow

The portfolio equations of land purchase and borrowings are similarly specified as:

$$ \left(\frac{IL_{t}}{K_{t-1}}\right)=\beta_{0}+\beta_{1}(AVQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\tau_{t})p_{t}}+\beta_{2}\frac{p_{t}^{L}L_{t-1}}{p_{t}^{I}K_{t-1}}+\beta_{3}\frac{CF_{\mathrm{t}}}{p_{t}^{I}K_{t-1}} $$

$$ \left(\frac{\Delta B_{t}}{p_{t}^{I}K_{t-1}}\right)=\gamma_{0}+\gamma_{1}(AVQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\tau_{t})p_{t}}+\gamma_{2}\frac{p_{t}^{L}L_{t-1}}{p_{t}^{I}K_{t-1}}+\gamma_{3}\frac{CF_{\mathrm{t}}}{p_{t}^{I}K_{t-1}} $$

In an alternative model( Model 2) the MAQ variable replaces the AVQ variable in eqs.(17) to (19). It should be noted that emphasis is not laid upon the role of stock market to convey the relevant information on the prospects of investment projects in Model 2.

The estimation results of Model 1 and 2 are shown in Table 11 and 12, respectively. $ ^{[20]} $ In Table 11 the AVQ variable, a key variable in investment equation, exerts a significantly negative effect on physical investment across all the industries except six industries (construction, textiles, fabricated metal products, machinery, electrical machinery, and wholesale and retail trade), which contradicts the verdict of the celebrated Tobin's q theory. Moreover, even when the AVQ variable exerts a significantly positive effect on investment, the explanatory power is low. The adjusted R-squared ranges from 0.0526 (textiles) to 0.4390 (electrical machinery). The AVQ variables also have a
the following manner.

Suppose that the market value of firm $ (V_{t}) $ is composed of not only the fundamentals but also the non-fundamentals such as bubbles or fads. That is, eq.(11) is rewritten as:

$$ (20)\;V_{t}=E_{t}\Biggl[\sum_{j=0}^{\infty}\beta_{t+j}D_{t+j}\Biggr]+N F_{t} $$

## where $ NF_{t} $ : non-fundamental parts of market valuation

It is easy to show that eq.(15) is still valid even when the market value of firm is contaminated by the non-fundamentals. $ ^{25} $ We add the following non-fundamentals term to eq.(15) in order to examine the possible effects of the non-fundamentals on investment behavior.

$$ (21)\left(\frac{I_{_{t}}}{K_{_{t-1}}}\right)=\alpha_{_{0}}+\alpha_{_{1}}(MAQ_{_{t}}-1)\frac{(1-z_{_{t}})p_{_{t}}^{^{I}}}{(1-\tau_{_{t}})p_{_{t}}}+\alpha_{_{2}}\frac{NF_{_{t}}}{(I-\delta)(I-\tau_{_{t}})p_{_{t}}K_{_{t-1}}} $$

The coefficients of $ \alpha_{1} $ and $ \alpha_{2} $ will measure the different impacts of the fundamentals and the nonfundamentals on investment, respectively. When $ \alpha_{2} $ is zero, the non-fundamentals will have no effect on capital investment. When $ \alpha_{1} $ is equal to $ \alpha_{2} $ , the non-fundamentals parts are as important as the fundamental parts. In this case the investment function is boiled down to eq.(16) since the following relationship between the average q defined by eq.(3) and the fundamentals is held:

$$ AVQ_{t}=MAQ_{t}+\frac{NF_{t}}{(1-\delta)(1-z_{t})p_{t}^{I}K_{t-1}} $$

In estimation the $ NF_{t} $ variable is constructed from eq.(22).

Eq. $ (21) $ is also extended by adding the land value and the cash flow of the firm to the list of explanatory variables. This is Model 3 and is expressed by:

$$ \left(\frac{I_{t}}{K_{t-1}}\right)=\alpha_{0}+\alpha_{1}(MAQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\tau_{t})p_{t}}+\alpha_{2}\frac{p_{t}^{L}L_{t-1}}{p_{t}^{I}K_{t-1}}+\alpha_{3}\frac{CF_{t}}{p_{t}^{I}K_{t-1}}+\alpha_{4}\frac{NF_{t}}{(1-\delta)(1-\tau_{t})p_{t}K_{t-1}} $$

$$ \left(\frac{IL_{t}}{K_{t-1}}\right)=\beta_{0}+\beta_{1}(MAQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\tau_{t})p_{t}}+\beta_{2}\frac{p_{t}^{L}L_{t-1}}{p_{t}^{I}K_{t-1}}+\beta_{3}\frac{CF_{t}}{p_{t}^{I}K_{t-1}}+\beta_{4}\frac{NF_{t}}{(1-\delta)(1-\tau_{t})p_{t}K_{t-1}} $$
$$ \left(\frac{\Delta B_{t}}{p_{t}^{I}K_{t-1}}\right)=\gamma_{0}+\gamma_{1}(MAQ_{t}-1)\frac{(1-z_{t})p_{t}^{I}}{(1-\tau_{t})p_{t}}+\gamma_{2}\frac{p_{t}^{L}L_{t-1}}{p_{t}^{I}K_{t-1}}+\gamma_{3}\frac{CF_{t}}{p_{t}^{I}K_{t-1}}+\gamma_{4}\frac{NF_{t}}{(1-\delta)(1-\tau_{t})p_{t}K_{t-1}} $$

The estimation results of Model 3 are shown in Table 13. There are interesting findings behind the results. First, the effects of the fundamentals and the non-fundamental parts are clearly different. The fundamentals exert a significantly positive effect on investment except a few cases, as is expected.

The fundamentals also have significantly positive effects on real land purchase and net borrowings for most of the cases. The effects of the non-fundamentals on the portfolio behavior are quite contrasted with those of the fundamentals. The effects of the non-fundamental parts on capital investment are negative for all the industries but textiles, steel, machinery, electrical machinery and wholesale and retail trade. Moreover they are significant at the 1% level for most of the cases. As was demonstrated in section 4, there persisted a positive overvaluation in the stock market during the middle to the latter half of 80's. If we interpret these positive persistence as the existence of bubbles or fads, then the evidence shows that they have negative effects on the real economy. This is quite reasonable, since the very fixty of the physical production facilities makes the sunk costs very high, which deters the investment behavior. $ ^{26)} $

Negative effects of the non-fundamentals are also observed for the land purchase and net borrowings for most of the industries. The non-fundamentals exert a significantly positive effect on borrowings for real estate.

Second, compared with Model 2, we find stronger support for a collateral role of land in the portfolio allocation of the corporations. Now the effect of the market value of land on investment is significantly positive at the 5% level for all industries, manufacturing, food and beverages, pulp, paper, and paper products, fabricated metal products, and wholesale and retail trade, while the effect of the market value of land on borrowings is significantly positive at the 5% level for all industries, pulp, paper, and paper products, fabricated metal products, electrical machinery, and real estate. These evidence support the view that the land value lessened the agency costs and thus increased the loans from banks and investment. However, the market value of land does not exert a positive effect on land purchase in a significantly manner, the reason being that the positive effect of land as a collateral on land purchase might be wiped out by a negative effect of the stock adjustment type.

As for the nature of collateral, it is often argued that the effects of the collateralizable net worth on investment and borrowings are not symmetric in the phase of business cycles. $ ^{[27]} $ In booming periods an increase of the net worth leads to that of investment and borrowings. However once the optimal level of investment without informational friction is attained, then investment will not increase.