Dynamic Structural Econometrics - Continuous Choice

GMM Estimation of Hansen and Singleton (1982)

Intro

Model

Consumption‑based Asset‑Pricing Model

Later, we will parameterize the utility function as for , but we will leave it in the general form for now.

• is the price of the asset in time t,
• is the quantity of the asset held by the consumer,
• is the dividend paid by the asset, and
• is the wage income of the consumer.

Taking the derivative with respect to and gives the following first-order conditions.

where is the real gross return on the asset in period t.This equation is the starting point for estimating the model.

Economic interpretation of the Euler equation

Solutions to economic models involving maximization of consumer utility subject to a budget constraint often take the form of a marginal rate of substitution (MRS) equal to a price ratio, and the Euler equation above has a similar intuition. The term is the (intertemporal) marginal rate of substition from period t to period , and the return on the asset for period t is , giving rise to a familiar interpretation of the Euler equation we use in estimation.

Estimation

Define the period t moment condition as

Then, for a set of instruments that is included in the information set known to the agent at time t (i.e., ), we have moment conditions defined by

.

• the discount factor β, and
• the risk aversion parameter γ, which defines the utility function

Defining , the theoretical model tells us that

The estimation procedure calculates sample moments by taking the sample average of across all observations and chooses a parameter vector θ to minimize the deviation of these sample moments from zero to as closely align the theoretical model with the data as possible. We go through the full procedure step-by-step in the coding blocks below.

Data requirements: We need data on consumption and asset returns to estimate the model. We obtain these from macroeconomic time series data.

Empirical implementation

1. Consumption data (FRED): real personal consumption expenditures (series name PCECC96)
2. CPI (FRED): consumer price index to convert nominal returns to real returns (series name CPIAUCSL)
3. Value-weighted equity returns (WRDS): value-weighted total market index returns with dividends (CRSP series name: VWRETD)
4. Equal-weighted equity returns (WRDS): equally-weighted total market index returns with dividends (CRSP series name: EWRETD)

Note on sample consutrction: I load the data from local files below. Without the WRDS/CRSP data, you will need to use different assets in the moment conditions. The variable toggles whether the WRDS/CSRP data is used from a local file stored in a "Data" subfolder or if all data is downloaded from FRED. If , then the implementation uses the following:

• NASDAQ price-only returns for equities (this is the variable defined below, which would otherwise be value-weighted returns).
• Risk-free returns as the second asset (this is the variable, which would otherwise be equal-weighted returns).

If you do not have any data downloaded, then change to in the code block below before running the code.

1. Quarterly data
2. Real gross returns
3. Per-capita consumption growth

% Change folder paths if your CSVs live elsewhere

datapath = 'Data\';

 

csvfile = @(fname) fullfile(datapath,fname);

 

% Sample end date (start date depends on data used, see below)

endDate = datetime(2019,12,31);

 

% Newey‑West bandwidth (quarters)

NWlag = 4;

 

%% Read raw CSVs

 

readLocal = true;

 

if readLocal == true

startDate = datetime(1960,1,1); % WRDS data goes back to 1960

 

pce = readtable(csvfile('PCECC96.csv'));

bill_raw = readtable(csvfile('DTB3.csv'));

cpi_raw = readtable(csvfile('CPIAUCSL.csv'));

vwr_raw = readtable(csvfile('crsp_agg_monthly_total_returns.csv'));

ewr_raw = vwr_raw; % same file, other column

 

 

vwr_raw = vwr_raw(:,{'DATE','vwretd'});

vwr_raw.Properties.VariableNames = {'observation_date','value'};

ewr_raw = ewr_raw(:,{'DATE','ewretd'});

ewr_raw.Properties.VariableNames = {'observation_date','value'};

else

% Read from the web directly

startDate = datetime(1972,1,1); % FRED NASDAQ data goes back to 1971

 

fredURL = @(series) sprintf(...

'https://fred.stlouisfed.org/graph/fredgraph.csv?id=%s',series);

 

url = fredURL('PCECC96');

tmp = [tempname,'.csv'];

websave(tmp, url);

pce = readtable(tmp);

 

url = fredURL('CPIAUCSL');

tmp = [tempname,'.csv'];

websave(tmp, url);

cpi_raw = readtable(tmp);

 

url = fredURL('DTB3');

tmp = [tempname,'.csv'];

websave(tmp, url);

bill_raw = readtable(tmp);

 

url = fredURL('NASDAQCOM');

tmp = [tempname,'.csv'];

websave(tmp, url);

vwr_raw = readtable(tmp);

vwr_raw.Properties.VariableNames(2) = {'value'};

vwr_raw.value = fillmissing(vwr_raw.value,'linear'); % fill missing values

 

end

 

 

%%% rename columns to generic names

pce.Properties.VariableNames(2) = {'value'};

bill_raw.Properties.VariableNames(2) = {'value'};

cpi_raw.Properties.VariableNames(2) = {'value'};

 

% fill missing values by linear interpolation

bill_raw.value = fillmissing(bill_raw.value,'linear');

 

 

%% Aggregate to quarterly

aggQ = @(tbl,fun) retime(table2timetable(tbl,'RowTimes','observation_date'), ...

'quarterly', fun, 'IncludedEdge','right');

 

pceQ = table2timetable(pce, 'RowTimes','observation_date');

billQ = aggQ(bill_raw, @mean);

cpiQ = aggQ(cpi_raw, @mean);

vwrQ = aggQ(vwr_raw, @(x) x(end)); % use end-of-period values

%vwrQ = aggQ(vwr_raw, @mean); % average within period

 

if readLocal == true

ewrQ = aggQ(ewr_raw, @(x) x(end)); % use end-of-period values

%ewrQ = aggQ(ewr_raw, @mean); % average within-period

 

vwrQ = lag(vwrQ,-1); % correct for timing mismatch

ewrQ = lag(ewrQ,-1);

% Merge & trim sample

Data = synchronize(pceQ, billQ, cpiQ, vwrQ, ewrQ);

Data = Data(timerange(startDate,endDate),:);

Data.Properties.VariableNames = {'pce','disc','cpi','vwr','ewr'};

 

% Convert to gross return

Data.vwr = 1 + Data.vwr;

Data.ewr = 1 + Data.ewr;

else

Data = synchronize(pceQ, billQ, cpiQ, vwrQ);

Data = Data(timerange(startDate,endDate),:);

Data.Properties.VariableNames = {'pce','disc','cpi','vwr'};

end

 

 

%% Build variables

logC = log(Data.pce);

gLead = diff(logC); % ΔlogC_{t+1}

disc = Data.disc(1:end-1)/100; % bill discount at t

Rf_nom = 1 ./ (1 - disc*90/360); % nominal bill return

pi_t = Data.cpi(2:end) ./ Data.cpi(1:end-1);

Rf_all = Rf_nom ./ pi_t; % real bill return

gLead = 1 + gLead; % C_{t+1}/C_t

n = length(gLead)-1;

gLag = gLead(1:n);

gC = gLead(2:end);

Rf = Rf_all(2:end);

if readLocal == true

Re_all = Data.ewr(2:end) ./ pi_t; % real equally‑weighted

Rv_all = Data.vwr(2:end) ./ pi_t; % real value‑weighted

else

Re_all = Rf_all; % Use risk-free rate as second asset

Rv_all = (Data.vwr(2:end)./Data.vwr(1:end-1)) ./ pi_t; %use NASDAQ price-only returns as first asset

end

Rv = Rv_all(2:end);

RvLag = Rv_all(1:n);

Re = Re_all(2:end);

ReLag = Re_all(1:n);

 

figure(1)

plot(Data.observation_date(2:end), gLead)

xlabel("Time")

ylabel("C_{t+1}/C_t")

title("Quarterly Consumption Growth")

if readLocal==true

title_str = "Quarterly Real Value-Weighted Returns";

else

title_str = "Quarterly Price-only NASDAQ Returns";

end

figure(2)

plot(Data.observation_date(2:end), Rv_all)

xlabel("Time")

ylabel("(P_{t+1} + D_{t+1})/P_t")

title(title_str)

 

if readLocal==true

title_str = "Quarterly Real Equal-Weighted Returns";

else

title_str = "Quarterly Risk-free Returns";

end

 

figure(3)

plot(Data.observation_date(2:end), Re_all)

xlabel("Time")

ylabel("(P_{t+1} + D_{t+1})/P_t")

title(title_str)

 

,

where and .

GMM Moment Conditions

Choose an instrument vector observable at time t.

.

Two‑step GMM Estimator

.

Minimise to obtain .

where K is the total number of moments. Recall that for our baseline estimation.

starting from . The result is , the two-step GMM estimate of θ.

.

Output is displayed below. We show the point estimates and standard errors for β and γ and the test of the overidentifying restrictions. Note that Hansen and Singleton use .

Functions

Newey-West weights

Moment function