Bullet point summary of lectures Highlights of articles
LECTURE 1
Introduction
Ideally, we would like an experiment but almost always we only have observational data (since
oftentimes these are “invisible” experiments). So, instead we rely on observational data.
Observational data is collected by e.g. government agencies or commercial entities.
In this course, we deal with difficulties arising from using observational data to estimate causal
effects, such as
- confounding effects (omitted factors)
- simultaneous causality
- “correlation does not imply causation”
Importantly: how do we measure and compare entities? Our choice of measure/parameters has
critical impact on our result and decisions —> qualify this clearly and well, and realize that other
measures may find different outcomes but that given our research, our choice is carefully chosen.
Econometrics itself is a precise and exact science, i.e. it is the only truth-mathematics. However,
when dealing with complex phenomena, econometrics is not always exact since mathematics
deals with numbers which are exact whereas econometrics deals with human beings who do not
behave consistently (across time). Econometrics does oftentimes give us a “general” conclusion.
Precision vs accuracy: ideally we prefer both to be high, but that is rather impossible due to
heterogenetic nature of humans (differing preferences). We always attempt to have highly
accurate estimates which are not necessary very precise. Accuracy is more important than
precision in research, as we do not wish to be precisely right about the wrong thing, it helps us
nothing. This may happen when you look for correlations which may happen at random and
happen to be precise but are not backed by any theoretical understanding (= inaccurate). This is
by no means helpful and we have to avoid those issues.
Questions slides 43:
- Air-disasters 2014-2016: The number of air-disasters actually decreased, but due to a few high-
profile disasters it made it seem like air-travel became more dangerous and since air-disasters
have become so rare, a disaster feels more than we are used to.
- Very many combinations for the first password, and little for the second one (common
password) and third one (if they know your name).
- Insurance benefits depends on the probability and costs you assign to damaging. For a phone:
high probability can make it worthwhile. For a house, even with a low probability it is
worthwhile given the costs of replacement.
- Insurance companies sells us these products since the chances are low and well diversified in a
large portfolio. Insurance market is essentially a game in probability.
Probability theory
A population can be infinitely large but can also be narrowly defined; depends on our entities of
interest. A random variable is neither random nor a variable. It is simply a numerical summary of
a random outcome.
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There are three approaches to assessing the probability of an uncertain event
- Classical probability: Assumes all outcomes in the sample space are equally likely to occur. It
requires a count of the outcomes in the sample space, relative to the number of outcomes that
satisfy a certain event.
- Relative frequency probability: an experiment carried out multiple times (= events = n) and you
measure the number of events that satisfy a certain event of interest (= Na) relative to total
number of events (= n). Or: number of events in the population that satisfy event A / total number
of events in the population.
- Subjective probability: An individual’s opinion/belief of a probability.
Statistical independence: if P(A n B) = P(A) x P(B), which relates to conditional probability,
since P(A | B) is then equal to P(A) —> thus statistically independent.
Probability distribution relates to how probabilities are distributed over a number of events (= n).
For example, when tossing coins multiple times and we are interested in some event happening
you cannot simply use a classical probability model (efficiently) but we need a probability
distribution.
Mean: expected value of Y , or the long-run average value of Y over repeated realizations of Y.
Variance: calculated as the expected value - actual observation, it is the measure of the squared
spread of the distribution.
Skewness and kurtosis: in this course we only talk about these in the context of
descriptive/qualitative statistical distribution.
Skewness measures the asymmetry of a distribution.
- (Zero) Skewness = 0 —> distribution is symmetric.
- Positive Skewness / Right Skewed, Skewness > 0, distribution has a right tail.
“Mean is to the right of median”. “Median is to the right of mode”.
- Negative Skewness / Left Skewed, Skewness < 0, distribution has a left tail.
“Mean is to the left of the median”.
“Median is to the left of mode”.
- The higher the skewness is different from zero, the more asymmetric the distribution.
Kurtosis: a measure of mass in tails, a measure of probability of large values, a measure of how
peaked a distribution is.
Kurtosis = 3: normal distribution.,
Kurtosis > 3: heavy tails (“leptokurtotic”).
- The higher the kurtosis is, the more peaked the distribution is and thus the more centred the
observations lie around the mean/median (so also shorter confidence intervals). But also a high
kurtosis would indicate heavier tails, or outliers, and vice versa for low kurtosis distributions —>
wider confidence intervals
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Correlation is a measure of how two variables are linearly associated with each other. The key
word here is “linear”. We cannot calculate a correlation coefficient for quadratic associations, so
correlation = 0.
When assessing a number/value, we can only make interpretations in relation to a
benchmark/average so it depends on the context in which we are speaking so it depends on the
average we are referring to.
A normal distribution:
- Bell-shaped and symmetrical.
- Skewness = 0, kurtosis = 3.
- Location is determined by mean, spread is determined by STDEV. Mean = median = mode.
- The random variable has an infinite theoretical range.
- Changes in the mean shits the distribution left of right. Changes in the STDEV increases or
decreases the spread.
- Any normal distribution can be transformed into a standardized normal distribution with a mean
of 0 and STDEV of 1 by computing it into Z-values as: (X - mean) / STDEV. The reason for
doing so is to better compare samples (e.g. different ranges or units of measurements, collected
from different entities/parts). An observation with a Z-value of 2 means: this observation is two
standard deviations (or two increments of one STDEV) above the mean. In a similar way, we
assess/interpret any value/number in relation to the mean and standard deviation.
- As the sample size/ number of observations increases, a distribution becomes more and more
normal. We should always try to get a large enough sample so that we could have a normal
distribution.
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LECTURE 2
Correlation only indicates linear associations, whereas inferential statistical can pick up both
linear and non-linear associations. Econometrics look at observational data and make inferences
and ultimately policy implications. We need solid theoretical understanding to make valid
inferences. We need to think about alternative explanations as well. We have to use econometrics
to not only make inferences about what we see but also to rule out alternate possibilities.
A hypothesis (for the mean) is set up to test theories and research questions and takes it into a
testable statement.
A p-value is the probability that the sample mean (or another statistic in another test) would be
greater than or equal to the actual observed results, assuming the null is true.
1% significance level two-sided, t = 2.575, so t-values > 2.575 are significant at 1% level
5% significance level two-sided, t = 1.96, so t-values > 1.96 are significant at 5% level
10% significance level two-sided (so 5% each side), t = 1.645, so t-values > 1.645 are significant
at 10% level. “we can say with 90% certainty that that the parameter is not equal to zero”.
P-value indicates whether the observed mean is in the rejection region (so lower than a certain
pre-specified significance level).
A p-value of 0.19 means that we would expect to find a sample mean of ... or smaller (when H1
<) in 19% of our samples, if the true population mean is ...
Note that the size of the P value for a coefficient says nothing about the size of the effect that
variable is having on your dependent variable - it is possible to have a highly significant result
(very small P-value) for a miniscule effect.
p-value: the probability that we would observe a more extreme test statistic in the direction of the
alternative than the one we observed. So low p-values mean low probabilities of our statistic —>
significant.
The smaller the significance level, the more precise the estimates are.
A confidence interval is calculated using the mean and STDEV: a 95% interval is the interval that
contains the true value of the (sample) mean in 95% of the repeated samples.
In any hypothesis test, we need distributional properties of our sample to check our sampled
entities, we need a valid sample to make proper inferences - random selection is key. If we reject
the null, the alternate hypothesis is true.
H1 < or H1 > , means one-sided test. So in the case of H1: U y < …, then the null is only rejected
if the sample mean is small enough, we do so with the t-values and p-values.
A t-statistic is calculated similarly to z-statistic except for that a t-value takes into account the
distributional properties of (small) samples.
Y-bar = sample mean
Uy = population mean
Sy = sample standard deviation
n = sample size
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