Regime switching and wages in major league baseball under the reserve clause

Abstract

Over the course of the twentieth century, American wages increased by a factor of about 100, while the wages of professional baseball players increased by a factor of 450, but that increase was neither smooth nor consistent. We use a unique and expansive dataset of salaries and performance variables of Major League Baseball pitchers that spans over 400 players and 60 years during the reserve clause era to identify factors that determine salaries and examine how the importance of various factors have changed over time. We employ a Markov regime-switching regression model borrowed from the macroeconomics literature, which allows regression coefficients to switch exogenously between two or more values as time progresses. This method lets us identify changes in wage determination that may have occurred because of a change in the league’s competitiveness, a change in the relative bargaining power between players and teams, or other factors that may be unknown or unobservable. We find that even though Major League Baseball was a tightly controlled monopsony with the reserve clause, there was a significant shift in salary determination that lasted from the Great Depression until after World War II where players’ salaries were more highly linked to their recent performance.

Appendices

Appendix 1: Filtering procedure

Hamilton (1989) describes an iterative procedure to evaluate a likelihood function for Markov regime switching for a single time series. In this appendix, we describe how we extend his method to a pooled panel regression model. Consider the following pooled regression model with regime switching,

$$ y_{i,t} = x_{i,t}'\beta(s_t) + e_{i,t}, $$

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where subscript i denotes a given individual, subscript t denotes a given time period, x _i,t is a vector of explanatory variables that may include both variables that vary across time for an individual and variables that remain constant over time. The regime state is given by $s_t \in \{1,\ldots,S\}$, where S is the number of regimes. The vector of coefficients is given by β(s _t ) = β _k , if s _t  = k, and the error term is independently and identically normally distributed, e _i,t ∼ N[0, σ(s _t )], where the standard deviation is given by σ(s _t ) = σ _k , if s _t  = k. The regime state, s _t , evolves according to the Markov chain, $P(s_t=k\,|\, s_{t-1}=j, \Uppsi_{t-1}) = p_{jk},$ where p _jk denotes the probability the economy switches from state j to state k as time enters period t and is another parameter to be estimated along with the other regression parameters, and $\Uppsi_{t-1}$ simply denotes all information up through period t − 1.

Given the error term e _i,t is normally distributed, if s _t  = k was known, the probability density function for y _i,t is given by,

$$ f(y_{i,t}\,|\,s_t=k, \Uppsi_{t-1}) = \frac{1}{\sqrt{2\pi\sigma_k^2}} exp \left\{ - \frac{\left(y_{i,t} - x_{i,t}'\beta_k\right)^2}{2 \sigma_k^2} \right\}. $$

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Let $f(y_{t} \,|\, \Uppsi_{t-1})$ denote the joint unconditional density function for all observations of the dependent variable in time t, where y _t denotes the set of observations for every individual at period $t, y_t \equiv \{y_{1,t}, y_{2,t}, \ldots, y_{n_t,t}\},$ and n _t is the number of individual for which data are available at time t. Each iteration begins with the input $P(S_{t-1}=j | \Uppsi_{t-1})$ for every $j\in\{1,\ldots,S\}$ and has the output $P(S_t=k | \Uppsi_t)$, and the process requires an initial condition for P(S ₀ = j). The filtering procedure takes as given the parameters β _k , σ _k , and p _jk for all j, k. Maximum likelihood estimates for these parameters can be obtained by maximizing the joint density function for all the data (the output from the filtering procedure) with respect to these parameters. The filtering algorithm follows these steps:

• Step 1: Find probabilities for being in each regime in time t, given information up through period t − 1. These probabilities are given by,

  $$ P(s_t=k\,|\,\Uppsi_{t-1}) = \sum_{j=0}^{S} P(s_t=k | s_{t-1}=j) P(s_{t-1}=j | \Uppsi_{t-1}), $$

  where P(s _t  = k | s _t-1 = j) ≡ p _jk is the Markov switching parameter, and $P(s_{t-1}=j | \Uppsi_{t-1})$ is known from the previous iteration (or initial condition).

• Step 2: Evaluate the conditional joint density function $f(y_t \,|\, \Uppsi_{t-1})$ which is computed by evaluating the following successive densities:

  $$ \begin{array}{l} f(y_t \,|\, s_t=k, \Uppsi_{t-1}) = \prod\limits_{i=1}^{n_t} f(y_{i,t} \,|\, s_t=k, \Uppsi_{t-1}), \\ f(y_t \,|\, \Uppsi_{t-1}) =\sum\limits_{k=1}^{S} f(y_t \,|\, s_t=k, \Uppsi_{t-1}) P(s_t=k \,|\, \Uppsi_{t-1}). \end{array} $$

  The first equation is valid since e _i,t and e _i′,t are independent for i ≠ i′, and the density $f(y_{i,t} \,|\, s_t=k, \Uppsi_{t-1})$ is given in Eq. 5. In the second equation, $P(s_t=k \,|\, \Uppsi_{t-1})$ is given from step 1.

• Step 3: Evaluate the updated probability for being in each regime in time t, given information up through period t − 1. These probabilities are given by,

  $$ \begin{aligned} P(s_t=k\,|\,\Uppsi_{t}) &= P(s_t=k\,|\,y_t, \Uppsi_{t-1}) = \frac{f(y_t, s_t=k | \Uppsi_{t-1})}{f(y_t | \Uppsi_{t-1})} \\ & = \frac{f(y_t \,|\, s_t=k, \Uppsi_{t-1}) P(s_t=k | \Uppsi_{t-1})}{f(y_t|\Uppsi_{t-1})}, \end{aligned} $$

  where the densities and probability needed to evaluate the second line are given in steps 1 and 2.

• Step 4: Return to step 1 until t = T, where T is the number of periods in the sample. The joint distribution for all the data is given by,

  $$ f(y^T | \Uppsi_{T-1}) = \prod_{t=1}^{T} f(y_t \,|\, \Uppsi_{t-1}), $$

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  where $f(y_t \,|\, \Uppsi_{t-1})$ is given from step 2. Taking logs, this can be transformed to the log-likelihood function,

  $$ l(y^T) = \sum_{t=1}^{T} \log \left( f(y_t \,|\, \Uppsi_{t-1}) \right). $$

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  Numerical maximization methods can be used to maximize Eq. 7 to obtain maximum likelihood estimates for β(s _t ) and σ²(s _t ) and transition probabilities p _j,k .

Appendix 2: Smoothing procedure

Once estimates for β(s _t ), σ²(s _t ) and all the transition probabilities are obtained, one may use the results from the filtering method to obtain smoothed estimates for $P(s_t=j | \Uppsi_T),$ the expected probability of being in each state for every period in the sample, using all the information from the sample. The smoothing procedure described here is unchanged from Hamilton (1989) and is described again here for convenience.

The smoothing procedure begins at the end of the sample period, and each iteration computes $P(s_t=k|\Uppsi_T)$ as its output from period t = T − 1 to t = 1, taking the output of the previous iteration, $P(s_{t+1}=l|\Uppsi_T),$ as an input. The starting value, $P(s_T=k | \Uppsi_T)$ is given from the output of Step 3 in the filtering procedure above for time t = T.

• Step 1: Compute conditional density $P(s_t=k | s_{t+1}=l, \Uppsi_t)$ based on output from the filtering procedure:

  $$ P(s_t=k | s_{t+1}=l, \Uppsi_t) = \frac{P(s_t=k, s_{t+1}=l | \Uppsi_t)}{P(s_{t+1}=l | \Uppsi_t)} = \frac{P(s_t=k, | \Uppsi_t) P(s_{t+1}=l | s_t=k)}{P(s_{t+1}=l | \Uppsi_t)}. $$

  Both $P(s_{t+1}=l | \Uppsi_t)$ and $P(s_{t}=k | \Uppsi_t)$ in the last expression are known from Step 1 of the filtering procedure and P(s _t+1 = l|s _t  = k) is the known Markov transition probability.

• Step 2: Approximate the full information joint density $P(s_t=k, s_{t+1}=l | \Uppsi_T)$ according to,

  $$ \begin{aligned} P(s_t=k, s_{t+1}=l | \Uppsi_T) &= P(s_{t+1}=l | \Uppsi_T) P(s_t=k | s_{t+1}=l, \Uppsi_T) \\ & \approx P(s_{t+1}=l | \Uppsi_T) P(s_t=k | s_{t+1}=l, \Uppsi_t). \end{aligned} $$

  In the second expression, $P(s_{t+1}=l | \Uppsi_T)$ is known from the previous iteration of the loop (or the initial condition) and $P(s_t=k | s_{t+1}=l, \Uppsi_t)$ is the output from Step 1.

• Step 3: The unconditional density $P(s_t=k|\Uppsi_T)$ is given by,

  $$ P(s_t=k|\Uppsi_T) = \sum_{l=1}^{S} P(s_t=k, s_{t+1}=l | \Uppsi_T). $$

• Step 4: Return to Step 1 until t = 1.