# RDP 2021-11: Smells Like Animal Spirits: The Effect of Corporate Sentiment on Investment 2. The Tobin's Q Model with Sentiment Shocks

I motivate my empirical approach to identifying the effect of sentiment on investment by outlining a simple extension to a basic Tobin's Q model. The outline closely follows the descriptions in Blundell et al (1992) and Bond and Cummins (2001).

A representative company optimally chooses investment to maximise the present value of the stream of current and expected future profits. The firm's objective is to maximise:

$V t = E t ∑ s=0 ∞ β t+s Π t+s$

where the expected valuation (Vt) of the company is a function of profits in each period $\left(\text{\hspace{0.17em}}{\Pi }_{t+s}\text{\hspace{0.17em}}\right)$ and a discount factor $\left(\text{\hspace{0.17em}}{\beta }_{t+s}\text{\hspace{0.17em}}\right)$. Corporate profits are assumed to have the form:

$Π t = p t Y( K t , N t )− w t N t − p t K [ I t +G( I t , K t ) ]$

where profits in the current period equal the difference between revenue and costs. Revenue is a function of production, which in turn depends on both the capital stock (Kt) and labour (Nt). Costs consist of both labour and capital costs, including capital adjustment costs (G(It,Kt)). The capital stock evolves according to the law of motion:

$K t+s = I t+s +( 1−δ ) K t+s−1$

where the capital stock at the end of period t + s (Kt+s) depends on investment during the period t + s (It+s) as well as the capital stock outstanding from the end of the previous period t + s − 1 (Kt+s−1).

The firm chooses investment to maximise the company's expected valuation subject to the capital stock law of motion. The first order conditions (FOCs) for investment are:

$λ t =− ∂ Π t ∂ I t λ t = E t [ ∑ s=0 ∞ β t+s ( 1−δ ) s ( ∂ Π t+s ∂ K t+s ) ]$

where the shadow value of an additional unit of capital is ${\lambda }_{t}$. Given price-taking behaviour, the first of the FOCs can be written:

$q t −1 ∂ G t ∂ I t$

Here, marginal q (qt) is equal to the ratio of the shadow value of an additional unit of capital to its purchase cost $\left(\text{\hspace{0.17em}}{\lambda }_{t}/{p}_{t}^{K}\text{\hspace{0.17em}}\right)$. Under certain conditions (such as constant returns to scale and price taking), marginal q can be approximated by average q:

$q t = V t p t K ( 1−δ ) K t−1$

$G( I t , K t )= b 2 [ ( I t K t )−c− e t ] 2 K t$

where adjustment costs depend on the investment rate, a fixed cost component and an idiosyncratic adjustment cost shock ( et ). From this, the Tobin's Q model of investment can be derived:

$( I t K t ) =c+ 1 b ( V t p t k ( 1−δ ) K t−1 −1 )+ e t =c + b 1 ( q t −1 )+ e t$

In which the error term (et) is the adjustment cost shock, observed by the firm but not by the researcher. Adjustment cost shocks are assumed to be white noise (although the key empirical results are not affected if we assume there is serial correlation in the error term).

Now assume that the company owners (or managers) can have different expectations to the market about the future value of the company. Suppose, in particular, that the company's expected valuation, which is equivalent to the valuation of the company's managers (Vt) is equal to the valuation of investors (as measured by the stock market) $\left(\text{\hspace{0.17em}}{\text{Ψ}}_{t}\text{\hspace{0.17em}}\right)$ and company-specific sentiment or noise shocks $\left(\text{\hspace{0.17em}}{\mu }_{t}\text{\hspace{0.17em}}\right)$ that have a mean of zero and variance of ${\sigma }_{\mu }^{2}$ :

$V t = Ψ t + μ t$

Here, I assume that the managers of the company can observe the share price and factor this into their own valuation. The managers' valuation of the company is also affected by sentiment shocks that reflect changes in manager expectations that are exogenous to company fundamentals. (This assumes a certain timing structure of information that will be discussed in more detail later.)

Alternative explanations for a difference in the expected valuations of managers and investors are possible. For example, the managers may know more about the fundamentals of the company than shareholders, in which case there will be some fundamental component to the sentiment shocks. This would imply that the sentiment shocks are not pure noise. I test for the possibility of private knowledge in an extension to the baseline model. Alternatively, managers and investors may have different discount factors $\left(\text{\hspace{0.17em}}{\beta }_{t+s}\text{\hspace{0.17em}}\right)$ with which they value future profits. For instance, managers may have extrapolative expectations about future investment, and the weight they place on future profits differs from those of investors (Gennaioli, Ma and Shleifer 2016).

This assumption implies that the average q ratio can be written:

$q t = Ψ t + μ t p t k ( 1−δ ) K t−1$

The average q ratio is a function of the market equity value (as measured by the stock market) $\left({\text{ψ}}_{t}\right)$ and sentiment shocks ( st ):

$q t = ψ t + μ t p t k ( 1−δ ) K t−1 = ψ t + s t$

The following model can therefore be estimated:

(1) $( I t K t )=c+ 1 b ( ψ t −1 )+ 1 b ( s t )+ e t$

where the rate of investment is a function of a standard measure of Tobin's Q (based on equity values) and corporate sentiment shocks. Equation (1) forms the baseline model which I take to the data and test the relationship between corporate investment, sentiment and fundamentals.