Ordinary Least Squares (OLS) is commonly used to estimate relationships between variables using observational data in economics. OLS finds the line of best fit by minimizing the sum of squared residuals to estimate parameters. The OLS estimator is a random variable that depends on the sample data. Asymptotically, as the sample size increases, the OLS estimator becomes consistent and its variance decreases. OLS provides the best linear unbiased estimates under the assumptions of the linear regression model.
Brief notes on heteroscedasticity, very helpful for those who are bigners to econometrics. i thought this course to the students of BS economics, these notes include all the necessary proofs.
Brief notes on heteroscedasticity, very helpful for those who are bigners to econometrics. i thought this course to the students of BS economics, these notes include all the necessary proofs.
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
The midpoint method or technique is a “measurement” and as each measurement it has a tolerance, but
worst of all it can be invalid, called Out-of-Control or OoC. The core of all midpoint methods is the accurate
measurement of the difference of the squared distances of two points to the “polar” of their midpoint
with respect to the conic. When this measurement is valid, it also measures the difference of the squared
distances of these points to the conic, although it may be inaccurate, called Out-of-Accuracy or OoA. The
primary condition is the necessary and sufficient condition that a measurement is valid. It is comletely
new and it can be checked ultra fast and before the actual measurement starts. .
Modeling an incremental algorithm, shows that the curve must be subdivided into “piecewise monotonic”
sections, the start point must be optimal, and it explains that the 2D-incremental method can find, locally,
the global Least Square Distance. Locally means that there are at most three candidate points for a given
monotonic direction; therefore the 2D-midpoint method has, locally, at most three measurements.
When all the possible measurements are invalid, the midpoint method cannot be applied, and in that case
the ultra fast “OoC-rule” selects the candidate point. This guarantees, for the first time, a 100% stable,
ultra-fast, berserkless midpoint algorithm, which can be easily transformed to hardware. The new algorithm
is on average (26.5±5)% faster than Mathematica, using the same resolution and tested using 42
different conics. Both programs are completely written in Mathematica and only ContourPlot[] has been
replaced with a module to generate the grid-points, drawn with Mathematica’s
Graphics[Line{gridpoints}] function.
La introducción de la incertidumbre en modelos epidemiológicos es un área de incipiente actividad en la actualidad. En la mayor parte de los enfoques adoptados se asume un comportamiento gaussiano en la formulación de dichos modelos a través de la perturbación de los parámetros vía el proceso de Wiener o movimiento browiniano u otro proceso discretizado equivalente.
En esta conferencia se expone un método alternativo de introducir la incertidumbre en modelos de tipo epidemiológico que permite considerar patrones no necesariamente normales o gaussianos. Con el enfoque adoptado se determinará en contextos epidemiológicos que tienen un gran número de aplicaciones, la primera función de densidad de probabilidad del proceso estocástico solución. Esto permite la determinación exacta de la respuesta media y su variabilidad, así como la construcción de predicciones probabilísticas con intervalos de confianza sin necesidad de recurrir a aproximaciones asintóticas, a veces de difícil legitimación. El enfoque adoptado también permite determinar la distribución probabilística de parámetros que tienen gran importancia para los epidemiólogos, incluyendo la distribución del tiempo hasta que un cierto número de infectados permanecen en la población, lo cual, por ejemplo, permite tener información probabilística para declarar el estado de epidemia o pandemia de una determinada enfermedad. Finalmente, se expondrá algunos de los retos computacionales inmediatos a los que se enfrenta la técnica expuesta.
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem, and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
Message: @Pi_vendor_247 on telegram if u want to sell PI COINS.
1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
Poonawalla Fincorp and IndusInd Bank Introduce New Co-Branded Credit Cardnickysharmasucks
The unveiling of the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card marks a notable milestone in the Indian financial landscape, showcasing a successful partnership between two leading institutions, Poonawalla Fincorp and IndusInd Bank. This co-branded credit card not only offers users a plethora of benefits but also reflects a commitment to innovation and adaptation. With a focus on providing value-driven and customer-centric solutions, this launch represents more than just a new product—it signifies a step towards redefining the banking experience for millions. Promising convenience, rewards, and a touch of luxury in everyday financial transactions, this collaboration aims to cater to the evolving needs of customers and set new standards in the industry.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
What website can I sell pi coins securely.DOT TECH
Currently there are no website or exchange that allow buying or selling of pi coins..
But you can still easily sell pi coins, by reselling it to exchanges/crypto whales interested in holding thousands of pi coins before the mainnet launch.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and resell to these crypto whales and holders of pi..
This is because pi network is not doing any pre-sale. The only way exchanges can get pi is by buying from miners and pi merchants stands in between the miners and the exchanges.
How can I sell my pi coins?
Selling pi coins is really easy, but first you need to migrate to mainnet wallet before you can do that. I will leave the telegram contact of my personal pi merchant to trade with.
Tele-gram.
@Pi_vendor_247
when will pi network coin be available on crypto exchange.DOT TECH
There is no set date for when Pi coins will enter the market.
However, the developers are working hard to get them released as soon as possible.
Once they are available, users will be able to exchange other cryptocurrencies for Pi coins on designated exchanges.
But for now the only way to sell your pi coins is through verified pi vendor.
Here is the telegram contact of my personal pi vendor
@Pi_vendor_247
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
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Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
If you are looking for a pi coin investor. Then look no further because I have the right one he is a pi vendor (he buy and resell to whales in China). I met him on a crypto conference and ever since I and my friends have sold more than 10k pi coins to him And he bought all and still want more. I will drop his telegram handle below just send him a message.
@Pi_vendor_247
Latino Buying Power - May 2024 Presentation for Latino CaucusDanay Escanaverino
Unlock the potential of Latino Buying Power with this in-depth SlideShare presentation. Explore how the Latino consumer market is transforming the American economy, driven by their significant buying power, entrepreneurial contributions, and growing influence across various sectors.
**Key Sections Covered:**
1. **Economic Impact:** Understand the profound economic impact of Latino consumers on the U.S. economy. Discover how their increasing purchasing power is fueling growth in key industries and contributing to national economic prosperity.
2. **Buying Power:** Dive into detailed analyses of Latino buying power, including its growth trends, key drivers, and projections for the future. Learn how this influential group’s spending habits are shaping market dynamics and creating opportunities for businesses.
3. **Entrepreneurial Contributions:** Explore the entrepreneurial spirit within the Latino community. Examine how Latino-owned businesses are thriving and contributing to job creation, innovation, and economic diversification.
4. **Workforce Statistics:** Gain insights into the role of Latino workers in the American labor market. Review statistics on employment rates, occupational distribution, and the economic contributions of Latino professionals across various industries.
5. **Media Consumption:** Understand the media consumption habits of Latino audiences. Discover their preferences for digital platforms, television, radio, and social media. Learn how these consumption patterns are influencing advertising strategies and media content.
6. **Education:** Examine the educational achievements and challenges within the Latino community. Review statistics on enrollment, graduation rates, and fields of study. Understand the implications of education on economic mobility and workforce readiness.
7. **Home Ownership:** Explore trends in Latino home ownership. Understand the factors driving home buying decisions, the challenges faced by Latino homeowners, and the impact of home ownership on community stability and economic growth.
This SlideShare provides valuable insights for marketers, business owners, policymakers, and anyone interested in the economic influence of the Latino community. By understanding the various facets of Latino buying power, you can effectively engage with this dynamic and growing market segment.
Equip yourself with the knowledge to leverage Latino buying power, tap into their entrepreneurial spirit, and connect with their unique cultural and consumer preferences. Drive your business success by embracing the economic potential of Latino consumers.
**Keywords:** Latino buying power, economic impact, entrepreneurial contributions, workforce statistics, media consumption, education, home ownership, Latino market, Hispanic buying power, Latino purchasing power.
2. OLS
Motivation
Economics is (essentially) observational science
Theory provides discussion regarding the relationship between variables
–Example: Monetary policy and macroeconomic conditions
What?: Properties of OLS
Why?: Most commonly used estimation technique
How?: From simple to more complex
Outline
1. Simple (bivariate) linear regression
2. General framework for regression analysis
3. OLS estimator and its properties
4. CLS (OLS estimation subject to linear constraints)
5. Inference (Tests for linear constraints)
6. Prediction
1
4. OLS
Correlation Coe¢ cient
Intended to measure direction and closeness of linear association
Observations: fyt; xtgT
t=1
Data expressed in deviations from the (sample) mean:
ezt = zt z; z = T 1
TX
t=1
zt; z = y; x
Cov(y; x) = E (yx) E (y) E (x)
sxy = T 1
TX
t=1
exteyt
which depends on the units in which x and y are measured
Correlation coe¢ cient is a measure of linear association independent of units
r = T 1
TX
t=1
ext
sx
eyt
sy
=
sxy
sxsy
; sz =
v
u
u
tT 1
TX
t=1
ez2
t ; z = y; x
Limits: 1 r 1 (applying Cauchy-Schwarz inequality)
3
5. OLS
Caution
Fallacy: “Post hoc, ergo propter hoc”(after this, therefore because of this)
Correlation is not causation
Numerical and statistical signi…cance, may not mean nothing
Nonsense (spurious) correlation
Yule (1926):
–Death rate - Proportion of marriages in the Church of England (1866-1911)
–r = 0:95
–Ironic: To achieve immortality ! close the church!
A few more recent examples
4
11. OLS
Simple linear regression model
Economics as a remedy for nonsense (correlation does not indicate direction of dependence)
Take a stance:
yt = 1 + 2xt + ut
–Linear
–Dependent / independent
–Systematic / unpredictable
T observations, 2 unknowns
In…nite possible solutions
–Fit a line by eye
–Choose two pairs of observations and join them
–Minimize distance between y and predictable component
min
P
jutj !LAD
min
P
u2
t !OLS
10
14. OLS
Simple linear regression model
De…ne the sum of squares of the residuals (SSR) function as:
ST ( ) =
TX
t=1
(yt 1 2xt)2
Estimator: Formula for estimating unknown parameters
Estimate: Numerical value obtained when sample data is substituted in formula
The OLS estimator (b) minimizes ST ( ). FONC:
@ST ( )
@ 1 b
= 2
X
yt
b1
b2xt = 0
@ST ( )
@ 2 b
= 2
X
xt yt
b1
b2xt = 0
Two equations, two unknowns:
b1 = y b2x
b2 =
sxy
s2
x
= r
sy
sx
=
PT
t=1 exteyt
PT
t=1 ex2
t
13
15. OLS
Simple linear regression model
Properties:
–b1; b2 minimize SSR
–OLS line passes through the mean point (x; y)
–but yt
b1
b2xt are uncorrelated (in the sample) with xt
Figure 9: SSR
14
16. OLS
General Framework
Observational data fw1; w2; :::; wT g
Partition wt = (yt; xt) where yt 2 R, xt 2 Rk
Joint density: f(yt; xt; ), vector of unknown parameters
Conditional distribution: f(yt; xt; ) = f(yt jxt; 1)f(xt; 2); f(xt; 2) =
R 1
1 f(yt; xt; )dy
Regression analysis: statistical inferences on 1
Ignore f(xt; 2) provided 1 and 2 are “variation free”
y: ‘dependent’or ‘endogenous’variable. x: vector of ‘independent’or ‘exogenous’variables
Conditional mean: m (xt; 3). Conditional variance: g (xt; 4)
m (xt; 3) = E (yt jxt; 3) =
Z 1
1
yf(yj xt; 2)dy
g (xt; 4) =
Z 1
1
y2
f(yj xt; 2)dy [m (xt; 3)]2
ut: di¤erence between yt and conditional mean:
yt = m (xt; 3) + ut (1)
15
17. OLS
General Framework
Proposition 1 Properties of ut
1. E (ut jxt) = 0
2. E (ut) = 0
3. E [h(xt)ut] = 0 for any function h ( )
4. E (xtut) = 0
Proof. 1. By de…nition of ut and linearity of conditional expectations,
E (ut jxt) = E [yt m (xt) jxt]
= E [yt jxt] E [m(xt) jxt]
= m (xt) m (xt) = 0
2. By the law of iterated expectations and the …rst result,
E (ut) = E [E (ut jxt)] = E (0) = 0
3. By essentially the same argument,
E [h(xt)ut] = E [E [h(xt)ut jxt]]
= E [h(xt)E [ut jxt]]
= E [h(xt) 0] = 0
4. Follows from the third result setting h(xt) = xt:
16
18. OLS
General Framework
(1) + …rst result of Proposition 1: regression framework:
yt = m (xt; 3) + ut
E (ut jxt) = 0
Important: framework, not model: holds true by de…nition.
m ( ) and g ( ) can take any shape
If m ( ) is linear: Linear Regression Model (LRM).
m (xt; 3) = x0
t
Y
T 1
=
2
4
y1
...
yT
3
5 ; X
T k
=
2
4
x0
1
...
x0
T
3
5 =
2
4
x1;1 x1;k
... ... ...
xT;1 xT;k
3
5 ; u
T 1
=
2
4
u1
...
uT
3
5
17
19. OLS
Regression models
De…nition 1 The Linear Regression Model (LRM) is:
1. yt = x0
t + ut or Y = X + u
2. E (ut jxt) = 0
3. rank(X) = k or det (X0
X) 6= 0
4. E (utus) = 0 8t 6= s
De…nition 2 The Homoskedastic Linear Regression Model (HLRM) is the LRM plus
5. E u2
t jxt = 2
or E (uu0
jX) = 2
IT
De…nition 3 The Normal Linear Regression Model (NLRM) is the LRM plus
6. ut N 0; 2
18
20. OLS
De…nition of OLS Estimator
De…ne the sum of squares of the residuals (SSR) function as:
ST ( ) = (Y X )0
(Y X ) = Y 0
Y 2Y 0
X + 0
X0
X
The OLS estimator (b) minimizes ST ( ). FONC:
@ST ( )
@ b
= 2X0
Y + 2X0
Xb = 0
which yield normal equations X0
Y = X0
Xb.
Proposition 2 b = (X0
X) 1
(X0
Y ) is the arg minST ( )
Proof. Using normal equations: b = (X0
X) 1
(X0
Y ). SOSC:
@2
ST ( )
@ @ 0
b
= 2X0
X
then b is minimum as X0
X is p.d.m.
Important implications:
–b is a linear function of Y
–b is a random variable (function of X and Y )
–X0
X must be of full rank
19
21. OLS
Interpretation
De…ne least squares residuals
bu = Y Xb (2)
b2
= T 1
bu0
bu
Y = Xb + bu = PY + MY ; where P = X (X0
X)
1
X0
and M = I P
Proposition 3 Let A be an n r matrix of rank r. A matrix of the form P = A (A0
A) 1
A0
is
called a projection matrix and has the following properties:
i) P = P0
= P2
(Hence P is symmetric and idempotent)
ii) rank(P) = r
iii) Characteristic roots (eigenvalues) of P consist of r ones and n-r zeros
iv) If Z = Ac for some vector c, then PZ = Z (hence the word projection)
v) M = I P is also idempotent with rank n-r, eigenvalues consist of n-r ones and r zeros,
and if Z = Ac, then MZ= 0
vi) P can be written as G0
G, where GG0
= I, or as v1v0
1 + v2v0
2 + ::: + vrv0
r where vi is a vector
and r = rank(P)
20
22. OLS
Interpretation
Y = Xb + bu = PY + MY
Y
Col(X)
MY
PY
0
Figure 10: Orthogonal Decomposition of Y
21
23. OLS
The Mean of b
Proposition 4 In the LRM, E
h
b jX
i
= 0 and Eb =
Proof. By the previous results,
b = (X0
X)
1
X0
Y = (X0
X)
1
X0
(X + u)
= + (X0
X)
1
X0
u
Then
E
h
b jX
i
= E
h
(X0
X)
1
X0
u jX
i
= (X0
X)
1
X0
E (u jX)
= 0
Applying the law of iterated expectations, Eb = E
h
E b jX
i
=
22
24. OLS
The Variance of b
Proposition 5 In the HLRM, V b jX = 2
(X0
X) 1
and V b = 2
E
h
(X0
X) 1
i
Proof. Since b = (X0
X) 1
X0
u;
V b jX = E b b
0
jX
= E
h
(X0
X)
1
X0
uu0
X (X0
X)
1
jX
i
= (X0
X)
1
X0
E [uu0
jX] X (X0
X)
1
= 2
(X0
X)
1
Thus, V b = E
h
V b jX
i
+ V
h
E b jX
i
= 2
E
h
(X0
X) 1
i
Important features of V b jX = 2
(X0
X) 1
:
–Grows proportionally with 2
–Decreases with sample size
–Decreases with volatility of X
23
25. OLS
The Mean and Variance of b2
Proposition 6 In the LRM, b2
is biased.
Proof. We know that bu = MY . It is trivial to verify that bu = Mu. Then, b2
= T 1
bu0
bu =
T 1
u0
Mu: This implies that
E b2
jX = T 1
E [u0
Mu jX]
= T 1
trE [u0
Mu jX]
= T 1
E [tr(u0
Mu) jX]
= T 1
E [tr(Muu0
) jX]
= T 1 2
tr(M)
= 2
(T k) T 1
Applying the law of iterated expectations we obtain Eb2
= 2
(T k) T 1
Unbiased estimator: e2
= (T k) 1
bu0
bu.
Proposition 7 In the NLRM, Vb2
= T 2
2 (T k) 4
Important:
–With the exception of Proposition 7, normality is not required
–b2
is biased, but it is the MLE under normality and is consistent
–Variance of b and b2
depend on 2
. bV b = e2
(X0
X) 1
24
26. OLS
b is BLUE
Theorem 1 (Gauss-Markov) b is BLUE
Proof. Let A = (X0
X) 1
X0
, so b = AY . Consider any other linear estimator b = (A + C) Y .
Then,
E (b jX) = (X0
X)
1
X0
X + CX = (I CX)
For b to be unbiased we require CX = 0, then:
V (b jX) = E (A + C) uu0
(A + C)0
As (A + C) (A + C)0
= (X0
X) 1
+ CC0
, we obtain
V (b jX) = V b jX + 2
CC0
As CC0
is p.s.d. we have V (b jX) V b jX
Despite popularity, Gauss-Markov not very powerful
–Restricts quest to linear and unbiased estimators
–There may be “nonlinear”or biased estimator that can do better (lower MSE)
–OLS not BLUE when homoskedasticity is relaxed
25
27. OLS
Asymptotics I
Unbiasedness is not that useful in practice (frequentist perspective)
It is also not common in general contexts
Asymptotic theory: properties of estimators when sample size is in…nitely large
Cornerstones: LLN (consistency) and CLT (inference)
De…nition 4 (Convergence in probability) A sequence of real or vector valued random
variables fxtg is said to converge to x in probability if
lim
T!1
Pr (kxT xk > ") = 0 for any " > 0
We write xT
p
! x or p lim xT = x.
De…nition 5 (Convergence in mean square) fxtg converges to x in mean square if
lim
T!1
E (xT x)2
= 0
We write xT
M
! x.
De…nition 6 (Almost sure convergence) fxtg converges to x almost surely if
Pr
h
lim
T!1
xT = x
i
= 1
We write xT
a:s:
! x.
De…nition 7 The estimator bT of 0 is said to be a weakly consistent estimator if bT
p
! 0.
De…nition 8 The estimator bT of 0 is said to be a strongly consistent estimator if bT
a:s:
! 0.
26
28. OLS
Laws of Large Numbers and Consistency of b
Theorem 2 (WLLN1, Chebyshev) Let E (xt) = t, V (xt) = 2
t , Cov (xi; xj) = 0 8i 6= j.
If lim
T!1
1
T
PT
t=1
2
t M < 1, then
xT T
p
! 0
Theorem 3 (SLLN1, Kolmogorov) Let fxtg be independent with …nite variance V (xt) =
2
t < 1. If
P1
t=1
2
t
t2 < 1, then
xT T
a:s:
! 0
Assume that T 1
X0
X ! Q (invertible and nonstochastic)
b = (X0
X)
1
X0
u
= T 1
X0
X
1
T 1
X0
u
p
! 0
b is consistent: b p
!
27
29. OLS
Analysis of Variance (ANOVA)
Y = bY + bu
Y Y = bY Y + bu
Y Y
0
Y Y = bY Y
0
bY Y + 2 bY Y
0
bu + bu0
bu
but bY 0
bu = Y 0
PMY = 0 and Y
0
bu = Y {0
bu = 0. Thus
Y Y
0
Y Y = bY Y
0
bY Y + bu0
bu
This is called the ANOVA formula, often written as
TSS = ESS + SSR
R2
=
ESS
TSS
= 1
SSR
TSS
= 1
Y 0
MY
Y 0LY
L = IT T 1
{{0
. If regressors include constant, 0 R2
1.
28
30. OLS
Analysis of Variance (ANOVA)
Measures percentage of variance of Y accounted for in variation of bY .
Not “measure”or “goodness”of …t
Doesn’t explain anything
Not even clear if R2
has interpretation in terms of forecast performance
Model 1: yt = xt + ut Model 2: yt xt = xt + ut with = 1
Mathematically identical and yield same implications and forecasts
Yet reported R2
will di¤er greatly
Suppose ' 1. Second model: R2
' 0, First model can be arbitrarily close to one
R2
is increases as regressors are added. Theil proposed:
R
2
= 1
SSR
TSS
T
T k
= 1
e2
b2
y
Not used that much today, as better model evaluation criteria have been developed
29
31. OLS
OLS Estimator of a Subset of
Partition X = X1 X2 = 1
2
Then X0
Xb = X0
Y can be written as:
X0
1X1
b1 + X0
1X2
b2 = X0
1Y (3a)
X0
2X1
b1 + X0
2X2
b2 = X0
2Y (3b)
Solving for b2 and reinserting in (3a) we obtain
b1 = (X0
1M2X1)
1
X0
1M2Y
b2 = (X0
2M1X2)
1
X0
2M1Y
where Mi = I Pi = I Xi (X0
iXi) 1
X0
i (for i = 1; 2).
Theorem 4 (Frisch-Waugh-Lovell) b2 and bu can be computed using the following algorithm:
1. Regress Y on X1; obtain residual eY
2. Regress X2 on X1; obtain residual eX2
3. Regress eY on eX2, obtain b2 and residuals bu
FWL was used to speed computation
30
32. OLS
Application of FWL: (Demeaning)
Partition X = X1 X2 where X1 = { and X2 is the matrix of observed regressors
eX2 = M1X2 = X2 { ({0
{)
1
{0
X2
= X2 X2
eY = M1Y = Y { ({0
{)
1
{0
Y
= Y Y
FWL states that b2 is OLS estimate from regression of eY on eX2
b2 =
TX
t=1
ex2tex0
2t
! 1 TX
t=1
ex2teyt
!
Thus the OLS estimator for the slope coe¢ cients is a regression with demeaned data.
31
33. OLS
Constrained Least Squares (CLS)
Assume the following constraint must hold:
Q0
= c (4)
Q (k q matrix of known constants), c (q-vector of known constants). q < k, rank(Q) = q.
CLS estimator of ( ) is value of that minimizes SSR subject to (4).
L ( ; ) = (Y X )0
(Y X ) + 2 0
(Q0
c)
is a q-vector of Lagrange multipliers. FONC:
@L
@ ;
= 2X0
Y + 2X0
X + 2Q = 0
@L
@ ;
= Q0
c = 0
= b (X0
X)
1
Q
h
Q0
(X0
X)
1
Q
i 1
Q0b c (5)
2
= T 1
Y X
0
Y X
is BLUE
32
34. OLS
Inference
Up to now, properties of estimators did not depend on the distribution of u
Consider the NLRM with ut N 0; 2
. Then:
yt jxt N x0
t ; 2
On the other hand as b = (X0
X) 1
X0
Y , then:
b jX
a
v N ; 2
(X0
X)
1
However, as b p
! , it also converges in distribution to a degenerate distribution
Thus, we require something more to conduct inference
Next, we discuss …nite (exact) and large sample distribution of estimators to test hypothesis
Components:
–Null hypothesis H0
–Alternative hypothesis H1
–Test statistic (one tail, two tails)
–Rejection region
–Conclusion
33
35. OLS
Inference with Linear Constraints (normality)
H0 : Q0
= c H1 : Q0
6= c
The t Test
q = 1. Assume u is normal, under the null hypothesis:
Q0b a
v N
h
c; 2
Q0
(X0
X)
1
Q
i
Q0b c
h
2Q0 (X0X) 1
Q
i1=2
N (0; 1) (6)
Test statistic used when is known. If not, recall
bu0
bu
2
2
T k (7)
As (6) and (7) are independent, hence:
tT =
Q0b c
h
e2
Q0 (X0X) 1
Q
i1=2
ST k
(6) holds even when normality of u is not present.
If H0 : 1 = 0, de…ne Q = 1 0 0
0
c = 0,
tT =
b1
q
bV1;1
34
36. OLS
Inference with Linear Constraints (normality)
Con…dence interval:
Pr bi z =2
q
bVi;i < i < bi + z =2
q
bVi;i = 1
Tail probability, or probability value (p-value) function
pT = p (tT ) = Pr (jZj jtT j) = 2 (1 (jtT j))
Reject the null when the p-value is less than or equal to
Con…dence interval for :
Pr
"
(T k) e2
2
T k;1 =2
< 2
<
(T k) e2
2
T k; =2
#
= 1 (8)
35
37. OLS
The F Test (normality)
q > 1. Under the null:
ST ST
b
2
2
q
When 2
is not known, replace 2
with e2
and obtain
ST ST
b
e2 =
T k
q
Q0b c
0 h
Q0
(X0
X) 1
Q
i 1
Q0b c
bu0bu
Fq;T k (9)
As with t tests, reject null when the value computed exceeds the critical value
36
38. OLS
Asymptotics II
How to conduct inference when u is not necessarilly normal?
Figure 11: Convergence in distribution
37
39. OLS
CLT
De…nition 9 (Convergence in distribution) fxtg is said to converge to x in distribution
if the distribution function FT of xT converges to the distribution F of x at every continuity
point of F. We write xT
D
! x and we call F the limiting distribution of fxtg. If fxtg and fytg
have the same limiting distribution, we write xT
LD
= yT .
Theorem 5 (CLT1, Lindeberg-Lévy) Let fxtg be i.i.d. with Ext = and Vxt = 2
. Then
ZT =
xT
[VxT ]1=2
=
p
T
xT D
! N (0; 1)
Assume that T 1
X0
X ! Q (invertible and nonstochastic) and that T 1=2
X0
u
D
! N 0; 2
Q
p
T b = T 1
X0
X
1
T 1=2
X0
u
D
! N 0; 2
Q 1
Thus, under the HLRM, asymptotic distribution does not depend on distribution of u
Normal vs t-test / Chi2
vs F test
38
40. OLS
Tests for Structural Breaks
Suppose we have two regimes regression
Y1 = X1 1 + u1
Y2 = X2 2 + u2
E
u1
u2
u0
1 u0
2 =
2
1IT1 0
0 2
2IT2
H0 : 1 = 2
Assume 1 = 2. De…ne
Y = X + u
Y =
Y1
Y2
, X =
X1 0
0 X2
, = 1
2
, and u =
u1
u2
Applying (9) we obtain:
T1 + T2 2k
k
b1
b2
0 h
(X0
1X1) 1
+ (X0
2X2) 1
i 1
b1
b2
Y 0
h
I X (X0
X)
1
X0
i
Y
Fk;T1+T2 2k (10)
where b1 = (X0
1X1) 1
X0
1Y1 and b2 = (X0
2X2) 1
X0
2Y2.
39
41. OLS
Tests for Structural Breaks
Same result can be derived as follows: De…ne SSR under alternative (structural change)
ST
b = Y 0
h
I X (X0
X)
1
X0
i
Y
and SSR under the null hypothesis
ST = Y 0
h
I X (X0
X)
1
X0
i
Y
T1 + T2 2k
k
ST ST
b
ST
b
Fk;T1+T2 2k (11)
Unbiased estimate of 2
is
e2
=
ST
T1 + T2 2k
Chow tests are popular, but modern practice is skeptic. Recent theoretical and empirical
applications: period of possible break as endogenous latent variable.
40
42. OLS
Prediction
Out-of-sample predictions for yp (for p > T) is not easy. In that period: yp = x0
p + up
Types of uncertainty:
–Unpredictable component
–Parameter uncertainty
–Uncertainty about xp
–Speci…cation uncertainty
Types of forecasts
–Point forecast
–Interval forecast
–Density forecast
Active area of research
41
43. OLS
Prediction
If HLRM holds, the predictor that minimizes MSE is bx0
p
bT
Given x, mean squared prediction error is
E
h
(byp yp)2
jxp
i
= 2
h
1 + x0
p (X0
X)
1
xp
i
To construct estimator of variance of forecast error, substitute e2
for 2
You may think that a con…dence interval forecast could be formulated as:
Pr byp z =2
q
bVbyp
< yp < byT+p + z =2
q
bVbyp
= 1
WRONG. Notice that
yp byp
r
2
h
1 + x0
p (X0X) 1
xp
i =
up + x0
p
b
r
2
h
1 + x0
p (X0X) 1
xp
i
Relation does not have a discernible limiting distribution (unless u is normal). We didn’t need
to impose normality for all the previous results (at least asymptotically).
We assumed that the econometrician knew xp. If x is stochastic and not known at T, MSE
could be seriously underestimated.
42
45. OLS
Measures of predictive accuracy of forecasting models
RMSE =
v
u
u
t 1
P
PX
p=1
(yp byp)2
MAE =
1
P
PX
p=1
jyp bypj
Theil U statistic:
U =
v
u
u
t
PP
p=1 (yp byp)2
PP
p=1 y2
p
U =
v
u
u
t
PP
p=1 ( yp byp)2
PP
p=1 ( yp)2
yp = yp yp 1 and byp = byp yp 1
or, in percentage changes,
yp =
yp yp 1
yp 1
and byp =
byp yp 1
yp 1
These measures will re‡ect the model’s ability to track turning points in the data
44
46. OLS
Evaluation
When comparing 2 models, is one model really better than the other?
Diebold-Mariano: Framework for comparing models
dp = g (bui;p) g (buj;p) ; DM =
d
p
V
d
! N (0; 1)
Harvey, Leyborne, and Newbold (HLN): Correct size distortions and use Student´s t
HLN = DM
P + 1 2h + h (h 1) =P
P
1=2
45
47. OLS
Finite Samples
Statistical properties of most methods: known only asymptotically
“Exact”…nite sample theory can rarely be used to interpret estimates or test statistics
Are theoretical properties reasonably good approximations for the problem at hand?
How to proceed in these cases?
Monte Carlo experiments and bootstrap
46
48. OLS
Monte Carlo Experiments
Often used to analyze …nite sample properties of estimators or test statistics
Quantities approximated by generating many pseudo-random realizations of stochastic process
and averaging them
–Model and estimators or tests associated with the model. Objective: assess small sample
properties.
–DGP: special case of model. Specify “true” values of parameters, laws of motion of
variables, and distributions of r.v.
–Experiment: replications or samples (J), generating arti…cial samples of data according
to DGP and calculating the estimates or test statistics of interest
–After J replications, we have equal number of estimators which are subjected to statistical
analysis
–Experiments may be performed by changing sample size, values of parameters, etc. Re-
sponse surfaces.
Monte Carlo experiments are random. Essential to perform enough replications so results
are su¢ ciently accurate. Critical values, etc.
47
49. OLS
Bootstrap Resampling
Bootstrap views observed sample as a population
Distribution function for this population is the EDF of the sample, and parameter estimates
based on the observed sample are treated as the actual model parameters
Conceptually: examine properties of estimators or test statistics in repeated samples drawn
from tangible data-sampling process that mimics actual DGP
Bootstrap do not represent exact …nite sample properties of estimators and test statistics
under actual DGP, but provides approximation that improves as size of observed sample
increases
Reasons for acceptance in recent years:
–Avoids most of strong distributional assumptions required in Monte Carlo
–Like Monte Carlo, it may be used to solve intractable estimation and inference problems
by computation rather than reliance on asymptotic approximations, which may be very
complicated in nonstandard problems
–Boostrap approximations are often equivalent to …rst-order asymptotic results, and may
dominate them in cases.
48