程序代写 Nonlinear Econometrics for Finance Lecture 1

Nonlinear Econometrics for Finance Lecture 1
. Econometrics for Finance Lecture 1 1 / 42

Welcome to

Nonlinear Econometrics for Finance!
. Econometrics for Finance Lecture 1 2 / 42

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
. Econometrics for Finance Lecture 1 3 / 42

Your instructor
. Professor
Web: https://sites.google.com/site/federicomariabandi/home
. Econometrics for Finance Lecture 1 4 / 42

Teaching Assistants (TAs)
1 Monday classes:
2 Tuesday classes:
3 Wednesday classes:
Rules of engagement:
For questions regarding Matlab and coding, please contact the TAs Make sure you are contacting the right TA for your class
All questions related to the course material should be directed to me
. Econometrics for Finance Lecture 1 5 / 42

1 Administrative details
2 Motivation – Intro to asset pricing
3 Review of methods
. Econometrics for Finance Lecture 1 6 / 42

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
. Econometrics for Finance Lecture 1 7 / 42

Administrative details
Course materials
Slides, lecture notes, books and software
All course materials will be available on OneDrive (there is a link on Blackboard in the “welcome announcement”)
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.)
. Econometrics for Finance Lecture 1 8 / 42

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 …
. Econometrics for Finance Lecture 1 9 / 42

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.
. Econometrics for Finance Lecture 1 10 / 42

Motivation – Intro to asset pricing
Fundamental principles of asset pricing
. Econometrics for Finance Lecture 1 11 / 42

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 …
. Econometrics for Finance Lecture 1 12 / 42

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.
. Econometrics for Finance Lecture 1 13 / 42
= 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).
. Econometrics for Finance Lecture 1 14 / 42

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.
. Econometrics for Finance Lecture 1 15 / 42

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. . Econometrics for Finance Lecture 1 16 / 42 Motivation - Intro to asset pricing The modern notion of the efficient markets hypothesis pt = Et [mt+1pt+1] , (14) where mt+1 is a random variable called a stochastic discount factor. Equivalently (we will return to the equivalence next week), pt = Et [mt+1pt+1] = EQt [pt+1] , (15) where we used risk-adjusted probabilities to compute the expectation. The role of the stochastic discount factor is to discount more riskier future cash flows. Equivalently, the risk-adjusted probabilities assign a higher probability to worse outcomes. In both cases, riskier assets are priced less. 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.) . Econometrics for Finance Lecture 1 17 / 42 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. . Econometrics for Finance Lecture 1 18 / 42 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 = 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 . Econometrics for Finance Lecture 1 20 / 42 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 understand deeply why pt = EQt [pt+1] = Et [mt+1(θ)pt+1] . 2 We will find mt+1(θ) for a fundamental asset pricing model in finance, the consumption Capital Asset Pricing Model. 3 We will discuss how the above moment condition can be used to estimate θ. . Econometrics for Finance Lecture 1 21 / 42 Review of methods Review of methods: asymptotic analysis . Econometrics for Finance Lecture 1 22 / 42 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 We will work with the sample mean but the same logic will extend to more complicated estimators, as we will see later in the course. . Econometrics for Finance Lecture 1 23 / 42 Review of methods Preliminary results . Econometrics for Finance Lecture 1 24 / 42 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 1􏰈T 1 =T μ=TTμ t=1 (20) =μ (21) = E(xt). (22) The sample mean XT is an unbiased estimator of the population . Econometrics for Finance Lecture 1 25 / 42 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. . 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 to infinity. If you combine the two properties above, you are effectively saying that the sample mean “goes” to μ as T → ∞. What does it mean to go to μ more formally? . Econometrics for Finance Lecture 1 27 / 42 Review of methods Consistency . Econometrics for Finance Lecture 1 28 / 42 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.
. 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.
. Econometrics for Finance Lecture 1 30 / 42

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.
. Econometrics for Finance Lecture 1 31 / 42

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!
. Econometrics for Finance Lecture 1 32 / 42

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). . Econometrics for Finance Lecture 1 33 / 42 Review of methods Asymptotic normality . Econometrics for Finance Lecture 1 34 / 42 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. . Econometrics for Finance Lecture 1 35 / 42 Review of methods CLT: alternative representations XT−μ Yd N(0,1) By simply standardizing ... √T􏰐X −μ􏰑Yd 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. . Econometrics for Finance Lecture 1 36 / 42 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 . Econometrics for Finance Lecture 1 37 / 42 Review of methods What about non i.i.d. samples? 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 . Econometrics for Finance Lecture 1 38 / 42 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. . Econometrics for Finance Lecture 1 39 / 42 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.
. Econometrics for Finance Lecture 1 40 / 42

Review of methods
Let us now see the WLLN and the CLT at work (Matlab code)
. Econometrics for Finance Lecture 1 41 / 42

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 OneDrive along with codes and data sets.
. Econometrics for Finance Lecture 1 42 / 42

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