Nonlinear Econometrics for Finance Lecture 1
Nonlinear Econometrics for Finance Lecture 1 1 / 43
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Welcome to
Nonlinear Econometrics for Finance!
Nonlinear Econometrics for Finance Lecture 1 2 / 43
The course in a nutshell
1 You are used (from Linear Econometrics for Finance) to estimating models written as follows:
yt = θ0 + θ1xt + εt, (1)
where xt is a predictor (it could be multivariate), {θ0,θ1} are parameters and εt is an error term. The function θ0 + θ1xt is, of course, linear.
2 After this course, you will be able to estimate (among many others) models written as follows:
yt =f(θ,xt)+εt, (2)
where xt is a predictor (it could be multivariate), θ defines a
parameter vector and εt is an error term. The function f(θ,xt) is,
in general, nonlinear. For example, f(θ,xt) = θ0xθ1. t
Nonlinear Econometrics for Finance Lecture 1 3 / 43
Your instructor
Associate Professor of Economics
Office Hours: Thursdays 11am-12:30pm Email:
Web: https://www.meleangelo.github.io
Nonlinear Econometrics for Finance Lecture 1 4 / 43
Teaching Assistants (TAs)
1 Wednesday sessions (84 and 85): )
2 Thursday sessions (82 and 83): Ronghao Lu
3 Friday sessions (81 and 86): )
Rules of engagement:
For questions regarding Matlab and coding, please contact the TAs Make sure you are contacting the right TA for your class
Nonlinear Econometrics for Finance Lecture 1 5 / 43
1 Administrative details
2 Motivation – Intro to asset pricing
3 Review of methods
Nonlinear Econometrics for Finance Lecture 1 6 / 43
Syllabus and grading
Administrative details
The syllabus is on Blackboard. Please read it carefully. Assignments and test:
3 homeworks, 15% of the final grade each, work in groups (maximum 3 people)
1 Due on week 3
2 Due on week 5
3 Due on week 7
Final exam, 55% of the final grade, cumulative
Nonlinear Econometrics for Finance Lecture 1 7 / 43
Administrative details
Course materials
Slides, lecture notes, books and software
All course materials will be available on Blackboard I will provide slides
I will provide lecture notes
I will provide academic articles, when needed
I suggest two optional books:
1 Econometrics, by (pdf available on OneDrive)
2 Asset Pricing, by (pdf available on OneDrive for an old,
but still very helpful, version)
3 We will not follow the books, but these are good readings if you are
interested in more detail and additional content
I will also provide sample codes in Matlab. (We will use Matlab heavily.)
Nonlinear Econometrics for Finance Lecture 1 8 / 43
Administrative details
An important observation
This is an advanced, research-based course.
You should approach it as your first-step into serious research.
If you do so, the payoff will be substantial irrespective of whether you remain in academia, and pursue a PhD, or join a firm in the private sector.
1 You should start reading academic articles (even beyond those that I will post online)
2 You should start questioning models and assumptions and use your creativity
3 You should be independent when it comes to all aspects of your work, including coding …
Nonlinear Econometrics for Finance Lecture 1 9 / 43
Administrative details
Regarding coding …
You will sometime have issues (everybody does).
When you do, you can ask the TA for your class, but only after:
1 consulting the Matlab help and/or the MathWorks forum
2 asking one (or more) of your peers (you will rarely code alone)
Again, this is a research-based course and you should begin training yourself to be creative and independent.
Nonlinear Econometrics for Finance Lecture 1 10 / 43
Class protocol
Administrative details
Please be so kind as to …
1 … keep your phones off.
2 … keep your cameras on (if on Zoom).
Both will lead to a superior learning community.
Nonlinear Econometrics for Finance Lecture 1 11 / 43
Motivation – Intro to asset pricing
Fundamental principles of asset pricing
Nonlinear Econometrics for Finance Lecture 1 12 / 43
Motivation – Intro to asset pricing
Efficient markets hypothesis
Suppose that an asset price at time t can be written as the expectation of some future “fundamental” value V ∗ conditional on information It available at time t:
pt = E[V ∗|It] = Et[V ∗] (3) The same equation has to hold tomorrow, i.e., at time t + 1:
pt+1 = E[V ∗|It+1] = Et+1[V ∗] (4) Note that It ⊂ It+1. In other words, the information set at time t is
contained in the information set at time t + 1 (people learn over time). Can we predict returns? No! Let’s see …
Nonlinear Econometrics for Finance Lecture 1 13 / 43
Motivation – Intro to asset pricing
Efficient markets hypothesis
Can we predict returns? No! Let’s see …
pt+1 −pt
Et[rt,t+1] = Et p (5)
p Et[pt+1 −pt]= p Et[Et+1[V ]−pt] (6)
p1 (Et[Et+1[V ∗]] − Et[pt]) t
= p1 (Et[V ∗] − pt) (7) t
By the law of iterated expectations (LIE), Et[Et+1[V ∗]] = Et[V ∗]. The expectation at time t of the expectation at time t + 1 is the same as the expectation at time t, because It ⊂ It+1. Intuition: folks cannot form expectations based on information they do not have yet.
In a frictionless market, prices contain all available information and
returns are unforecastable (because Et[rt,t+1] = 0) given the past.
Nonlinear Econometrics for Finance Lecture 1 14 / 43
= p1(pt−pt)=0. t
Motivation – Intro to asset pricing
Efficient markets hypothesis
Is it obvious that Et[rt,t+1] = 0 implies that returns are unforecastable? Let’s see …
Consider a predictor xt in the time t information set.
C[rt,t+1, xt] (9)
= E[rt,t+1 xt ] − E[rt,t+1 ]E[xt ] (10)
= E[Et [rt,t+1 xt ]] − E[Et [rt,t+1 ]]E[xt ] (11)
= E[xt Et [rt,t+1 ]] − E[Et [rt,t+1 ]]E[xt ] (12)
Because the covariance between returns and variables known at time t is zero, returns are unforecastable given time t information (the beta of a regression of rt,t+1 on xt would be zero).
Nonlinear Econometrics for Finance Lecture 1 15 / 43
Motivation – Intro to asset pricing
Efficient markets hypothesis
Another implication of Et[pt+1 − pt] = 0:
pt = Et[pt+1].
Prices are martingales: the best predictor for tomorrow’s price is today’s price.
Nonlinear Econometrics for Finance Lecture 1 16 / 43
Motivation – Intro to asset pricing
A simple example
Consider2numbers: uasin“up”sothatu>1anddasin“down”so that d < 1.
price now pt
price tomorrow if up
(pt+1 = upt)
(pt+1 = dpt) Figure: Binomial tree: price up or down with some probabilities.
price tomorrow if down
WehaveEt[pt+1]=pupupt+(1−pup)dpt =pt ifpupu+(1−pup)d=1 or if pup = 1−d . This is a restriction on the true probabilities to
guarantee that the price process is a martingale.
Nonlinear Econometrics for Finance Lecture 1 17 / 43
Motivation - Intro to asset pricing
The modern notion of the efficient markets hypothesis
pt = Et [mt+1pt+1] , (14) or, equivalently (we will return to the equivalence next week),
pt=EQt [pt+1], (15) where mt+1 is a random variable called a “stochastic discount factor”.
The role of the stochastic discount factor is to change the probabilities from true probabilities to risk-adjusted probabilities.
Thus, the price process is still a martingale. However, it is not a martingale with respect to the true probabilities. It is a martingale with respect to risk-adjusted probabilities. (Some folks call them “risk-neutral probabilities,” a term which makes - economically - little sense.)
Nonlinear Econometrics for Finance Lecture 1 18 / 43
Motivation - Intro to asset pricing
A simple example: with risk-adjusted probabilities
Consider2numbers: uasin“up”sothatu>1anddasin“down”so that d < 1.
price now pt
price tomorrow if up
(pt+1 = upt)
(pt+1 = dpt) Figure: Binomial tree: price up or down with some probabilities.
price tomorrow if down
We have EQt [pt+1] = pQupupt + (1 − pQup)dpt = pt if pQupu + (1 − pQup)d = 1 or if pQup = 1−d . This is a restriction on the risk-adjusted probabilities
to guarantee that the price process is a martingale. You will see (or
have seen) analogous expressions in Derivatives.
Nonlinear Econometrics for Finance Lecture 1 19 / 43
Motivation - Intro to asset pricing
True and risk-adjusted probabilities
Why are the Q-probabilities risk-adjusted? Higher probability given to worse outcomes.
price now pt
price tomorrow if up pup = 0.5
1−pup =0.5
price tomorrow if down
price tomorrow if up pQup = 0.4
1 − pQup = 0.6
price tomorrow if down
(pt+1 = upt)
(pt+1 = dpt)
(pt+1 = upt)
price now pt
(pt+1 = dpt) Nonlinear Econometrics for Finance Lecture 1
Motivation - Intro to asset pricing
Asset pricing models
1 Asset pricing models impose discipline on the stochastic discount factor, i.e., the risk-adjustment mt+1.
2 Different models imply a different mt+1.
3 In all cases, mt+1 can be parametrized. We can write mt+1(θ),
where θ is a scalar parameter or a vector of parameters to be
estimated.
4 In other words, we would have
pt = EQt [pt+1] = Et [mt+1(θ)pt+1] , (16)
5 Equivalently, dividing everything by pt, we can write
= Et [mt+1(θ)(1 + rt,t+1)] , (17)
but this is a moment condition which can be estimated to test a specific asset pricing model.
1 = Et mt+1(θ) p
Nonlinear Econometrics for Finance Lecture 1 21 / 43
Motivation - Intro to asset pricing
Asset pricing models
The moment condition
1 = Et [mt+1(θ)(1 + rt,t+1)] , (18)
is, in general, nonlinear in the parameter θ. Hence, the importance of this course.
1 Next week, we will find mt+1(θ) for a fundamental asset pricing model in finance, the consumption Capital Asset Pricing Model.
2 We will discuss how the above moment condition can be used to estimate θ.
Nonlinear Econometrics for Finance Lecture 1 22 / 43
Review of methods
Review of methods: asymptotic analysis
Nonlinear Econometrics for Finance Lecture 1 23 / 43
Review of methods
It is useful to consider what happens to estimators (like the least-squares estimator in your Linear Econometric for Finance classes) when the sample size grows
We will say: as T → ∞, where T is the number of observations
In this section, we will review two fundamental concepts of asymptotic analysis:
1 Consistency
2 Asymptotic normality
Nonlinear Econometrics for Finance Lecture 1 24 / 43
Review of methods
Preliminary results
Nonlinear Econometrics for Finance Lecture 1 25 / 43
Review of methods
Mean of i.i.d. samples
Let us consider a population with mean μ and variance σ2 < ∞. The probability distribution is generic.
Let X = {x1,x2,...,xT} be an i.i.d. sample of size T:
1 T 1 T
E(XT) = E T xt =T E(xt) (19)
t=1 t=1 1T 1
=T μ=TTμ t=1
(20) =μ (21)
= E(xt). (22) The sample mean XT is an unbiased estimator of the population
Nonlinear Econometrics for Finance Lecture 1 26 / 43
Review of methods
Variance of i.i.d. samples
Let us consider a population with mean μ and variance σ2 < ∞. The probability distribution is generic.
Let X = {x1,x2,...,xT} be an i.i.d. sample of size T:
1 2 σ2 = T2Tσ=T.
V(XT ) = V xt = T t=1
T2V(x1 +x2 +...+xT)
1 (V(x1)+...+V(xT)) T2
The larger the sample size T, the smaller the variance of XT. Nonlinear Econometrics for Finance Lecture 1
Review of methods
i.i.d. samples
Thus, the standard deviation of the sample mean is
σ SD(XT)=√ .
What do we learn from these computations?
The sample mean is unbiased for μ. It gives an estimate of the population mean that is correct “on average”.
The larger the sample size, the smaller the variance V(XT ) of the sample mean. Not only does the sample mean give us μ (on average), the distribution of the sample mean around μ becomes more and more concentrated as the number of observations increases without bound.
Both properties are very useful if we want to use the sample mean as an estimator of the population mean.
Nonlinear Econometrics for Finance Lecture 1 28 / 43
Review of methods
Consistency
Nonlinear Econometrics for Finance Lecture 1 29 / 43
Review of methods
Convergence in probability
Letg1,g2,g3,....,gT beasequenceofrandomvariables(forexample, 123T
xt/1, xt/2, xt/3, ..., xt/T). t=1 t=1 t=1 t=1
We say that gT converges in probability to a constant c as T →∞if,forallε>0,
We write or
lim Pr(|gT −c| > ε) = 0. T→∞
gT →p c, PlimgT =c.
Nonlinear Econometrics for Finance Lecture 1
Review of methods
Convergence in Mean-Squared
Letg1,g2,g3,….,gT beasequenceofrandomvariables.WesaythatgT converges in mean-squared to a constant c if
E(gT−c)2 →0. (28) T→∞
m.2. gT → c.
Important Result: convergence in mean-squared implies convergence in probability.
Nonlinear Econometrics for Finance Lecture 1 31 / 43
Review of methods
Convergence in probability of the sample mean
Let us return to our example of the sample mean of an i.i.d. sample XT . Notice that, as T → ∞,
V(XT)=E XT −E(XT) =E XT −μ = T →0. (29)
This means that the sample mean converges in mean-squared to μ: m.2.
which, in turn, implies convergence in probability
This is called the Weak Law of Large Numbers (WLLN): the sample mean converges in probability to the mean of the population as the number of observations increases.
Nonlinear Econometrics for Finance Lecture 1 32 / 43
Review of methods
WLLN and Consistency
Weak Law of Large Numbers (WLLN): The sample mean converges in probability to the mean of the population as the number
of observations increases:
We say that the sample mean XT is a consistent estimator for the
population mean μ as the number of observation increases.
This is the same as:
xt → E(xt).
Sample averages converge to the corresponding expectation!
Nonlinear Econometrics for Finance Lecture 1 33 / 43
What about non i.i.d. samples?
Review of methods
The sample mean is consistent for μ = E(xt), whatever μ is.
Again, more generally, sample averages converge to the corresponding expectation, under fairly general assumptions:
1 sufficiently low dependence in the data
2 boundedness of the second moment
In our example above, we assumed i.i.d. data (data with no dependence) with σ2 < ∞.
We will, thus, apply the result more generally (to stationary data, for example).
Nonlinear Econometrics for Finance Lecture 1 34 / 43
Review of methods
Asymptotic normality
Nonlinear Econometrics for Finance Lecture 1 35 / 43
Review of methods
Central Limit Theorem (CLT)
Let us consider a population with mean μ and variance σ2 < ∞. The probability distribution is generic.
Let X = {x1,x2,...,xT} be an i.i.d. sample of size T: We have shown that:
1 E(XT ) = μ
2 V(XT ) = σ2
The Central Limit Theorem says that averages are approximately normally distributed as T → ∞, that is:
XTYN μ,T , (30)
where the symbol Yd means approximately distributed as.
Nonlinear Econometrics for Finance Lecture 1 36 / 43
Review of methods
CLT: alternative representations
XT−μ Yd N(0,1)
By simply standardizing ...
√TX −μYd N(0,σ2) 2T
The sample mean converges to μ at a speed of convergence of T. This is, also, the speed of convergence to zero of the standard deviation of the sample mean.
Tt=1(xt−μ) Yd N(0,σ2)
Differences from the mean (suitably standardized) converge to a normal random variable.
Nonlinear Econometrics for Finance Lecture 1 37 / 43
Review of methods
On the third representation
We will make extensive use of the third way to state the CLT. Indeed, notice that
Tt=1(xt − μ) Tt=1 E(xt − μ)
E √ = √ =0. (31)
It makes sense for the limiting normal random variable to have mean 0. In fact, we are averaging de-meaned random variables.
For the variance, we have
Tt=1(xt − μ) Tt=1 V(xt − μ) Tσ2 2
V √T = T =T=σ. (32) Hence, we can also write:
T(x−μ) T(x−μ) t=1 t d t=1 t
√ YN0,V √ . (33) TT
Nonlinear Econometrics for Finance Lecture 1 38 / 43
What about non i.i.d. samples?
Review of methods
In our discussion above, we assumed an i.i.d. sample with σ2 < ∞. The result is, however, general.
Standardized (by
variables converge to normal random variables under fairly mild assumptions
sufficiently low dependence in the data (an i.i.d. sample has no dependence)
bounded second moments
We will therefore apply the result more generally (i.e., to stationary data).
T ) sample means of de-meaned random
Nonlinear Econometrics for Finance Lecture 1 39 / 43
Review of methods
WLLN vs CLT
The weak law of large numbers (WLLN) is about the convergence of sample means, i.e., objects like
Tt=1 xt T
to the corresponding expected value (μ = E(x)).
The central limit theorem (CLT) is about the convergence of
standardized (by
T ) sample means of de-meaned observations
Tt=1(xt − μ)
to zero-mean normal random variables.
Nonlinear Econometrics for Finance Lecture 1 40 / 43
Review of methods
Combining the WLLN and the CLT: Slutsky’s theorem
IfzT Yd zandcT →p casT→∞,then 1 zT+cTYdz+c.
2 zTcTYdzc.
3 zTYdzifc̸=0.
Example. Assume zT Yd N(0,σ2) (by the CLT) and cT is a consistent estimator of
the variance σ2 (by the WLLN). For example, assume cT is the empirical variance.
Then, √T Y N(0,σ ) = N(0,1), given 3, because σ > 0.
We will make use of these results in all proofs of convergence.
Nonlinear Econometrics for Finance Lecture 1 41 / 43
Review of methods
Let us now see the WLLN and the CLT at work (Matlab code)
Nonlinear Econometrics for Finance Lecture 1 42 / 43
Review of methods
For your review
In the lecture notes, there is a Chapter 0:
1 Chapter 0 contains a brief review of matrix algebra
2 Chapter 0 also contains a review of methods of asymptotic inference
(which we just discussed).
Please read Chapter 0 carefully before the next class. This material was covered in Linear Econometrics and I will assume you have working familiarity with it.
The first homework will give you a chance to review the material taught in Linear Econometrics.
In order to facilitate your work and help you with the review, you will find a set of lecture notes on Linear Econometrics on Blackboard along with codes and data sets.
Nonlinear Econometrics for Finance Lecture 1 43 / 43
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