Dynamic Structural Econometrics - CCP estimation

The random utility model

• Utility that is additively separable in observed state x and idiosyncratic utility term ϵ. Let denote the utility from selling the current car. We represent this utility as

• Normalizing the deterministic component of utility for one choice to zero. In this case, we let for all x so that .
• The idiosyncratic utility draws are iid from the same distribution: Type I Extreme Value. In particular, this means that follows a Logistic distribution, so that Pr, the familiar logistic probability.
• Finally, we assume that the utility from selling the car is linear in the observed mileage: , where represents the monetary benefit to the individual from not having to obtain a new, more reliable car and is the expected cost of repairs if the car breaks down before reaching the destination.

Suppose we observe a set of decisions from N individuals about whether or not to take the trip with the current car. If , the car was replaced for individual n, and if , then the car was kept.

Taking the log and plugging in and gives the log-likelihood associated with observation n:

Maximizing the log-likelihood with respect to the parameters θ conditional on the observed decisions and car mileages will generate the structural estimates. The code below simulates some sample data and estimates this model.

Moving from static to dynamic

Bus engine replacement

We assume that, as in Rust (1987), we have monthly data on bus maintenance records. This includes data on a set of buses over T months. The observations include mileage readings for each bus and and indicator for whether the engine has been replaced.

Model

• Time is discrete, with denoting each period, with .
• Each period, Harold Zurcher chooses between two actions. He can either replace the bus engine, , or not replace . These are mutually exclusive, so that .
• Zurcher's payoff depends on a single state variable that represents the mileage of the bus. Mileage advances by a random amount if the bus engine is not replaced and is set to zero if the bus engine is replaced. Denote by the distribution over next period mileage conditional on current mileage after making choice d. When the bus engine is replaced, mileage resets to zero, so that for all . Mileage accumulation when the engine is not replaced follows a discrete analog of the exponential distribution up to some maximum mileage where mileage increment is .

• In addition to mileage , there is a permanent and observable bus-specific type . Assume that the deterministic component of flow payoffs for each action for a bus of type s are given by

• Payoffs also have an additively separable idiosyncratic component so that the total flow payoff from choice j in period t is . We assume that are i.i.d draws from a Type I EV distribution.
• Let denote the time-varying state variable together with the permanent bus type s.
• Denoting the optimal decision rule at t by , we define the ex-ante value function for state in period t as

• The conditional value function is the value function for choice j in period t and state less the utility shock . In the case of Zurcher's bus engine replacement problem, these functions can be expressed as

where we have "undone" the combination of and s in the state variable to make clear that the state transition function f only affects mileage accumulation and not the permanent bus type s.

• Note that all t subscripts have been removed from value functions and state transitions: this is a stationary, infinite horizon problem.
• In the binary choice setting with Type I EV preference shocks we can express conditional choice probabilities very simply:

• Representation: finally, we can use the results of Hotz and Miller (1993) to express the difference between the value function and conditional value function :

where when Type I EV, where is the Euler-Mascheroni constant.

Representation and identification

The mapping between the difference conditional value functions and the conditional choice probabilities forms the basis for representation and identification. To generate the conditions that establish identification, we use the restrictions imposed on the utility function. In our case, we have assumed that for all states . Define as the distribution over state variables reached after periods with respect to the choice sequence . For simplicity, assume for now that there is only a single bus type s, so that we can effectively ignore this state variable and express states and transitions only in terms of bus mileage s. The distribution over period τ states is expressed as

In our setting, we will define , i.e., the reference choice in each period is to replace the engine. Repeated application of using replacement as the reference choice in each period generates that

This equation establishes identification of for each : everything on the right-hand side is known () or can be estimated (). It is useful to "stack" the moment condition equations for each state x and express the moment conditions in matrix form.

Two ways to develop an estimator.

When , we can use the matrix version of geometric series convergence to express the utility as:

Note that everything on the right-hand side is known (utilities, β, transition matrices) or can be estimated ().

Recall that if we can express in terms of the parameters θ, then we can use the logit estimator directly. While we can derive an expression directly from the representation identity, it is easiest to work from the unrestricted utility estimates above.

This form allows us to use the logit estimator: is a known function of the structural parameter vector θ, and everything else is known or can be estimated.

where ∗ is element-wise multiplication, is the vector of choice probabilities for choice d and λ is an vector of ones.

Methods 1 and 2 above are especially useful when both (i) is linear in θ and (ii) for all x. In settings without these assumptions, Method 3 using parameters from both utility functions and may be a more practical way to represent the value function. Note that the second term in parentheses above is simply the probability that each choice is optimal multiplied by the expected utility conditional on that choice being the optimal choice for each possible choice d.

Data

We have set and . With mileage starting at zero, there are 21 state values.

Knowing the state values is needed for calculating the flow payoffs , but for nearly every other calculation we are primarily interested in the state's index. We call this the encoded state: it maps value associated with a state to that state's number. For example, in the representation above, the encoded state of mileage = 0.2 is 5. (Note that this will change slightly below when we add the bus types to the state variable.)

Calculating the probability of transitioning from one state to another requires referring to the relevant row and column indices in the state transition matrix f. Recall that for . We can define the state transition matrix in two steps. First, we assign the probabilities for each state , and then we lump all of the leftover probability from mileage larger than into the last state.

Defined in this way, each row i of f corresponds to the probability distribution over states conditional on state i. The first row is the distribution over next period states conditional on the bus engine having a mileage of zero. The probability of remaining at zero is , the probability of mileage increasing to state 2 is , and so on. All elements below the diagonal are zero because the bus mileage cannot decrease without replacing the engine.

We use the following parameters and data settings to simulate the data, where π is the fraction of buses that have :

beta = 0.9;

theta = [2.00, -0.15, 1.0];

T=15;

S=1:2;

X=x;

N=1000;

pi=0.4;

[data_t, data_x, data_s, data_d, data_x_index] = generate_data(N, T, S, f, F, X,beta, pi, theta);

 

 

plot(data_t(1,1:12), data_x(1,1:12));

xline(data_t(1,data_d(1,1:12)==1),'--');

%xline(7,'--');

xlabel('Time'); ylabel('Mileage');title('Mileage and replacement for bus #1');

legend('Mileage','Engine replacement','location','northwest');

 

% Optional: save data

%Data_vars={'data_t', 'data_x', 'data_s', 'data_d', 'data_x_index'};

%Const_vars={'beta','theta','S','X','x_len','pi','T','x','N','f','F','F1','F2','emc'};

%All_vars=[Data_vars, Const_vars];

%save("Bus_engine_ccp_data.mat",All_vars{:});

CCP estimation

Remember, the goal at this stage is choice prediction using the observed states. Many different estimators can be used in this first stage. The primary objective is that capture how states predict choices as accurately as possible. We estimate λ using the parametric representation above and then use to generate fitted values for the conditional choice probabilities.

Combining states: to have fewer expressions to work with, we will combine the two states, mileage and bus type, into a single state variable () and work with this object.

• The variable is the encoded state of each observation in the data. It maps each observation to a row of the matrix.
• The matrix and are the combined state transitions for all states (all mileage states plus the two observed bus types). These are constructed by blocks of zeros and the original state transition matrices because bus types cannot change: a "type 2" bus will always be a "type 2" bus, so all probabilities associated with "type 1" bus states are zeros.

Recalling that , it is straightforward to define the log-likelihood function. This is similar to the log-likelihood for the static random utility model; the difference is that there is a state-specific offset term (our "BigV_diff") applied to each observation ("BigV_est").

Full-information maximum likelihood estimation

Exercise: using the unrestricted utility estimates directly

,

where , , is an vector of zeros, and is an vector of utilities for choice 2.

where and are vectors of the utilities associated with each state.

Exercise: Implement this estimator using the estimated conditional choice probabilities from the previous sections. Use the identity matrix as a weighting matrix. When you construct the estimator, use one moment condition for each observation; that is, you should use the variable to "copy" moment conditions corresponding to the number of times they appear in the data. You should estimate θ using an OLS estimator.

(Bonus: Method 3. Stationary representation of the ex-ante value function.)

where ∗ is element-wise multiplication, is the vector of choice probabilities for choice d and λ is an vector of ones. The relevant term for evaluating the logit likelihood is , which we can express as

with the above expression for , we can now have an expression for the differenced conditional value function that we can use in the logit likelihood.

To simplify the representation, we construct the components of the ex-ante value function in pieces. We start with the term , which we define as

Next, we define :

Finally, we have . Recalling that , we can express this as . We're going to leave off the θ term for now, as this will be the main argument in the optimization function:

Functions

Stationary CCP likelihood function

Data generation

Full-information maximum likelihood

Value function iteration

References