Introduction to Statistics and Econometrics · Probability and Statistics for Economists Introduction to Statistics and Econometrics May 23, 2019 16 Fundamental Axioms of Econometrics

Introduction to Statistics

and Econometrics

Professor Yongmiao Hong

Cornell University

May 23, 2019

May 23, 2019 2Introduction to Statistics and EconometricsProbability and Statistics for Economists

1.1 General methodology of modern economic research

1.2 Roles of Econometrics

1.3 Illustrative Examples

1.4 Roles of Probability and Statistics

CONTENTS

Introduction to Statistics and Econometrics May 23, 2019 3Probability and Statistics for Economists

Step 1: Data collections and summary of empirical stylized facts

Data collections:

surveys field studies experimental

economics Big data

General methodology of modern economic research Step 1: Data collections and summary of empirical stylized facts

Data collections

economic theories/models

Empirical validation/inference

Applications



The so-called stylized facts are often summarized from observed economic data.


Example 1：Engel curve in microeconomics

The share of a consumer’s expenditure on a commodity out of her or his total income will vary as his/her income changes;



Example 2: Phillips Curve in macroeconomics:

A negative correlation between the inflation rate and the unemployment rate in an aggregate economy;




Example 3: volatility clustering in finance:

A high volatility today tends to be followed by another high volatility tomorrow, a low volatility today tends to be followed by another low volatility tomorrow, and both alternate over time.





the empirical stylized facts

a starting point for economic research

serve as


Step 2: Development of economic theories/models

● With the empirical stylized facts in mind, economists then develop an economic theory or model.

● This usually calls for specifying a mathematical model of economic theory.

General methodology of modern economic research Step 2: Development of economic theories/models

Data collections



Applications


Step 2: Development of economic theories/models

• An example is the Euler equation for rational expectations in macroeconomics.

• The objective of economic modeling is not merely to explain the stylized facts, but also to understand the economic mechanism.

General methodology of modern economic research Step 2: Development of economic theories/models


Step 3: Empirical validation/inference of economic models

General methodology of modern economic research Step 3: Empirical validation/inference of economic models

• A key is to transform an economic model into a testable empirical econometric model.

• One often has to assume some functional form, up to some unknown model parameters, or to choose suitable instrumental variables to form a set of moment conditions.

Data collections



Applications


Step 3: Empirical validation/inference of economic models

General methodology of modern economic research Step 3: Empirical validation/inference of economic models

• One needs to estimate unknown model parameters and make inferences based on observed data.

• Check whether the econometric model is adequate. An adequate model should be at least consistent with the empirical stylized facts.


Step 4: Applications

After an econometric model passes the empirical evaluation, it can then be used to:

General methodology of modern economic research Step 4: Applications

Explain important empirical stylized facts

Test economic theory and/or hypotheses

Forecast future evolution of the economy

Policy evaluation and other applications

Data collections



Applications






CONTENTS


Fundamental Axioms of Econometrics

Modern econometrics is essentially built upon the following fundamental axioms:

Roles of Econometrics Fundamental Axioms of Econometrics

Any economy can be viewed as a stochastic process governed by some probability law.

Economic phenomenon, often summarized in form of data, can be reviewed as a realization of this stochastic data generating process (DGP).


Remarks:

● Modern economy is full of uncertainty, e.g., market (demand, supply and price) uncertainty, policy uncertainty

● The probability law of this stochastic economic system characterizes the evolution of the economy, and can be viewed as “law of economic motions.”






● The goal of econometrics:

To infer the probability law of the stochastic economic system based on observed data, and then use the inferred probability law for economic applications.

For example, economic theory usually takes a form of imposing certain restrictions on the probability law. Thus, one can test economic theory or economic hypotheses by checking the validity of these restrictions.




● Tools and methods of probability and statistics will provide operating principles for econometrics. For examples,

● Econometrics is not a simple application of a general theory of mathematical statistics to economic data.

model specification

parameter estimation

hypothesis testing

model validation




“Econometrics is by no means the same as economic statistics. Nor is it identical with what we call general economic theory, although a considerable portion of this theory has a definitely quantitative character. Nor should econometrics be taken as synonymous with the application of mathematics to economics. Experience has shown that each of these three viewpoints, that of statistics, economic theory, and mathematics, is a necessary, but not by itself a sufficient, condition for a real understanding of the quantitative relations in modern economic life. It is the unification of all three that is powerful. And it is this unification that constitutes econometrics.”

Ragnar Frisch (1933)


Among other things, econometrics can play the following roles in economics:

▫ Examine how well an economic theory can explain historical economic data (particularly the important empirical stylized facts);

▫ Test validity of economic theories and economic hypotheses;




▫ Predict the future evolution of the economy

▫ Recommend business strategies and evaluate economic policies.




● To appreciate the roles of modern econometrics in economic analysis, we now discuss a number of illustrative econometric examples in various fields of economics and finance.








CONTENTS


where

𝑌𝑡 is aggregate income,

𝐶𝑡 is private consumption,

𝐼𝑡 is private investment,

𝐺𝑡 is government spending,

𝜀𝑡 is consumption shock.

Illustrative Examples Example 1: Keynes Model, Multiplier and Policy Recommendation

Example 1: Keynesian Model, Multiplier and Policy Recommendation

The Simple Keynesian Model


The parameters 𝛼 and 𝛽 can have appealing economic interpretations:

• 𝛼 is survival level consumption,

• 𝛽 is the marginal propensity to consume (MPC).




Multiplier of income with respect to government spending:

which depends on the marginal propensity to consume 𝛽.




To assess the effect of fiscal policies on the economy, it is important to know the magnitude of 𝛽.




Suppose a representative agent has a constant relative risk aversion utility

where𝛽>0 is the agent's time discount factor,𝛾≥0 is the risk aversion parameter,𝜇(∙) is the agent's utility function in each time period,𝐶𝑡 is consumption during period 𝑡.

Illustrative Examples Example 2: Rational Expectations and Dynamic Asset Pricing Models

Example 2: Rational Expectations and Dynamic Asset Pricing Models


The agent's optimization problem is to choose a sequence of consumptions 𝐶𝑡 over time to

subject to the intertemporal budget constraint

where

𝑞𝑡 is the quantity of the asset purchased at time 𝑡,

𝑊𝑡 is the agent’s period 𝑡 income.




Define the marginal rate of intertemporalsubstitution

where model parameter vector 𝜃 = 𝛽, 𝛾 ′.




First order condition (FOC) of the agent's optimization problem:

where

𝑅𝑡+1 is the stochastic gross return on the risky asset,

𝐼𝑡 is information set available at time 𝑡.




● This FOC is usually called the Euler equation of the economic system.

● Questions:

● • How to estimate this model?

● • How to test validity of a rational expectations model?

● Need to use generalized method of moments (GMM) (see Hansen 1982).




Production function:

where

𝑌𝑖 = output of firm 𝑖,

𝐿𝑖 = labor of firm 𝑖,

𝐾𝑖 = capital of firm 𝑖,

𝜀𝑖 = is a shock (e.g., uncertain weather condition if

𝑌𝑖 is an agricultural product).

Illustrative Examples Example 3: Production Function and Hypothesis on Constant Return to Scale

Example 3: Production Function and Hypothesis on Constant Return to Scale


Constant return to scale (CRS):




CRS is a necessary condition for the existence of a long-run equilibrium of a competitive market economy.

• If CRS does not hold, and the technology displays the increasing return to scale (IRS), then the industry will lead to natural monopoly.




A conventional approach to testing CRS:

• Assume the production function is a Cobb-Douglas function:

• Then CRS becomes a mathematical restriction on parameters (𝛼, 𝛽) :

• If 𝛼 + 𝛽 > 1, the production technology displays IRS.




● In statistics, a popular procedure to test one-dimensional parameter restriction is Student's t-test.

● Unfortunately, this test is not suitable for many cross-sectional economic data, which usually display conditional heteroskedasticity.

● One needs to use a robust, heteroskedasticity-consistent test procedure, originally proposed in White (1980).




It should be emphasized that CRS is equivalent to the statistical hypothesis

under the assumption that the production technology is a Cobb-Douglas function. This additional condition on the production function is not part of CRS and is called an auxiliary assumption.




If the auxiliary assumption is incorrect, the statistical hypothesis

will not be equivalent to CRS. Correct model specification is essential for a valid conclusion and interpretation for econometric inference.




Extended Cobb-Dauglas production function (after taking a logarithmic operation)

where 𝑖 is the index for firm 𝑖 ∈ 1,… ,𝑁 and 𝑡 is the index for year𝑡 ∈ 1,… , 𝑇 ,

• Bonus𝑖𝑡 is proportion of bonus out of total wage bill,

• Contract𝑖𝑡 is proportion of workers who have signed a fixed-term contract.

Illustrative Examples Example 4: Effect of Economic Reforms on Transitional Economy

Example 4: Effect of Economic Reforms on Transitional Economy


● This is an example of the so-called panel data model (see, e.g., Hsiao 2003).

● Bonuses and fixed-term contracts were two innovative incentive reforms in Chinese state-owned enterprises in 1980s, compared to the fixed wage and life-time employment systems in the pre-reform era.

● To examine effects of these incentive reforms, we consider the null statistical hypothesis




● Conventional F-tests in classical linear regression would serve our purpose, if we can assume conditional homoskedasticity.

● Unfortunately, this cannot be used because there may exist the other way of causation from Y𝑖𝑡 to Bonus𝑖𝑡; namely a productive firm may pay its workers higher bonuses regardless of their efforts:




● Causality from bonus to productivity will cause correlation between bonuses and error term 𝑢𝑖𝑡, rendering the OLS estimator inconsistent and invalidating the conventional t-tests or F-tests.

● Fortunately, econometricians have developed an important estimation procedure called Instrumental Variables estimation, which can effectively filter out the impact of the causation from output to bonus and obtain a consistent estimator for the bonus parameter.




In evaluating the effect of economic reforms, we have turned an economic hypothesis into a statistical hypothesis




When the hypothesis

is not rejected, we should not conclude that reforms have no effect. This is because the extended production function model, where reforms are specified additively, is only one of many ways to check effect of reforms. For example, one could also specify a model such that the reforms affect marginal productivities of labor and capital.




Thus, when the hypothesis

is not rejected, we can only say that we do not find evidence against the economic hypothesis that the reforms have no effect.




Weak form of efficient market hypothesis (EMH):

It is impossible to predict future stock returns using the history of past stock returns:

where

𝑌𝑡 = asset return at time 𝑡,

𝐼𝑡−1 = {𝑌𝑡−1, … , 𝑌1} is the information set at time 𝑡 − 1.

Illustrative Examples Example 5: Efficient Market Hypothesis (EMH) and Predictability of Financial Returns

Example 5: Efficient Market Hypothesis (EMH) and Predictability of Financial Returns


● When EMH holds, past information of stock returns has no predictive power for future stock returns.

● An important implication of EMH is that mutual fund managers will have no informational advantage over layman investors.




Question: How to test EMH?

• One simple way to test EMH is to consider autoregression:

𝑌𝑡 = 𝛼0 +

𝑗=1

𝑝

𝛼𝑗𝑌𝑡−𝑗 + 𝜀𝑡

where

p is a pre-selected number of lags,

𝜀𝑡 is a random disturbance.




Question: How to test EMH?

● EMH implies

Thus, any nonzero coefficient 𝛼𝑗, 1 ≤ 𝑗 ≤ 𝑝, is evidence

against EMH.

● One can test whether the 𝛼𝑗 are jointly zero. The

classical F-test can be used to test 𝐇0 when 𝑣𝑎𝑟(𝜀𝑡 𝐼𝑡−1 = 𝜎2, i.e., there exists conditional homoscedasticity.




● However, EMH may coexist with volatility clustering ( i.e., 𝑣𝑎𝑟(𝜀𝑡 𝐼𝑡−1 may be time-varying), which is one of the most important empirical stylized facts of financial markets. This implies that the standard F-test statistic cannot be used, even asymptotically. One has to use procedures that are robust to conditional heteroskedasticity.




● When one rejects the null hypothesis 𝐇0 that the 𝛼𝑗 are

jointly zero, we have evidence against EMH. However, when one fails to reject 𝐇0, one can only conclude that we do not find evidence against EMH. The reason is, again, that the linear AR(p) model is one of many possibilities to check EMH.




● Since 1970s, oil crisis, the floating foreign exchanges system, and the high interest rate policy in the U.S. have stimulated a lot of uncertainty in the world economy.

Illustrative Examples Example 6: Volatility Clustering and ARCH Models

Example 6: Volatility Clustering and ARCH Models


● Volatility is a key instrument for measuring uncertainty and risk in finance. This concept is important to investigate information flows and volatility spillover, financial contagions between financial markets, options pricing, and calculation of Value at Risk.




● Volatility can be measured by the conditional variance of asset return 𝑌𝑡 given the information 𝐼𝑡−1:




● A popular volatility model is the AutoRegressiveConditional Heteroskedasticity (ARCH) model, originally proposed by Engle (1982).



Engle's (1982) ARCH(q) model


● ARCH(q) can explain a well-known stylized fact in financial markets-volatility clustering. It can also explain the non-Gaussian heavy tail of asset returns.




● Question: How to estimate a volatility model?

• Because the conditional distribution of 𝑌𝑡 is unknown, the popular maximum likelihood estimation (MLE) method cannot be used.

• Nevertheless, one can assume that standardized innovation {𝑧𝑡}is i.i.d.N(0,1), and obtain a conditional distribution of 𝑌𝑡 given 𝐼𝑡−1 Then one can estimate model parameters using MLE.




● Question: How to estimate a volatility model?

• Since {𝑧𝑡} is not necessarily i.i.d.N(0,1), the above estimation procedure is called Quasi-MLE or QMLE (White 1982).

• QMLE is consistent for true model parameters, but its asymptotic variance is larger than that of the MLE (i.e., when the true distribution of {𝑧𝑡} is known)








CONTENTS


Econometrics is an integration of statistical modeling and inferences of economic observational data with economic theory.

Roles of Probability and Statistics Roles of Probability and Statistics

Roles of Probability and Statistics

The objective of econometrics is to infer laws of economic motions of a stochastic economy

Theory, methods and tools of econometrics are mathematical statistics in combination with economic theory.


● Probability theory provides operating rules and understanding of mathematical statistics.

● In fact, as perhaps the best mathematical tool to describe uncertainty, probability theory has been very useful analytic tools for macroeconomics, microeconomics and finance.




● Probability and Statistics together provide a foundation for econometrics. In particular, they provides the concepts, tools and methods of mathematical statistics which are needed to understand econometrics.




Modern statistics usually

• assumes that a mathematical probability model generates an observed data.

• assumes that the probability model often contains some unknown parameters.

• based on the observed data, develop methods to estimate unknown parameters and statistical hypotheses.




For such statistical analysis, one would need to know a variety of important concepts and tools, such as:



• mean• variance• quantile • conditional mean• conditional variance• conditional quantile• correlation• sample and data

• maximum likelihood estimation• method of moment estimation• t- and F-distributions• convergence concepts

(e.g., almost sure convergence)? • law of large numbers• central limit theorems


As a distinctive feature from other probability / statistics courses, this course will offer intuitions, explanations and potential applications for tools and methods of mathematical statistics from an economic perspective.






How to characterize income inequality by cumulative probability distribution? Lorenz curve!

Why are mean and variance important in economic analysis?

What is an economic interpretation for law of large numbers?

What are economic implications of model misspecication? Model risk!


There are many different approaches of data analysis. In the era of Big Data, which contain structural and unstructured data, an emerging field for Big data analysis is data science, which employs different approaches (e.g., statistics, machine learning, etc.) to analyze data.




Statistics remains as an important approach to analyzing data including Big data.




Textbook & Website

Textbook

Websitehttps://probability.xmu.edu.cn

概率论与统计学中国统计出版社,2017

Probability and Statistics for Economists

World Scientific Publishing Company (November 2, 2017)

https://probability.xmu.edu.cn/


Thank You !

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Introduction to Statistics and Econometrics · Probability and Statistics for Economists Introduction to Statistics and Econometrics May 23, 2019 16 Fundamental Axioms of Econometrics