Productivity for Infonomos

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Productivity for Infonomos

• Presented for the IES-Network by
• Heiko Lampe

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Overview

• Just for Infonomos, not part of the official
  program
• Two sessions with unknown length
• Preparation of the basic mathematical and
  statistical concepts for Productivity
• Some repetition of old QM1 and QM2 stuff, many
  might have forgotten by now
• Acknowledged by Christian Kerckhoffs and Huub
  Meijers
• For free
• Dates
• Monday 22nd 16-18
• Wednesday 24th 13.30 ???

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Disclaimer

• All the explanations are meant as a supportive
  act. I try to be as correct as possible, but
  obviously mistakes can happen. I do not take any
  responsibilities for these mistakes. I cannot
  possibly teach you everything of 4 weeks QM3
  within 2 sessions! Nevertheless, I try my best.

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Agenda Meeting 1/1

• Easy stuff (math)
• - Derivatives and Leibniz (differential)
  notation
• - ln rules
• - Taylor approximation (1st order)
• - Elasticities and Eulers Theorem (Homogeneity)
• - Growth formula
• - Implicit differentiation

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Agenda Meeting 1/2

• Easy stuff (stats) Mostly intuition
• Unbiasedness of the OLS Estimator
• Omitted Variable Bias
• F-Test
• Indirect t-test (teta-trick)
• Harder stuff 1 and 2 (QM3)
• Discrete Dynamics and difference equations
• Continuous Dynamics and differential equations
• Easy stuff (QM3)
• Phase Diagrams and Equilibria

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Agenda Meeting 2

• All the stuff we did not make in the first
  meeting
• Lecture 1 for productivity from Kerkhoffs

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Easy stuff (math) 1/4

• Rules for derivatives
• Power rule
• Product rule
• Quotient rule
• Chain rule (important)
• Leibniz (differential) notation

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Easy stuff (math) 2/4

• Ln rules
• General rule
• Multiplication
• Division
• Taylor approximation (first order)
• f(x0) about a f(a) f (a) (x0- a)

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Easy stuff (math) 3/4

• Elasticities
• The El. of a multiplication is equal to the sum
  of the partial El.
• General
• Degree of homogeneity (k) from Eulers Theorem
  and alternative way (t-trick)

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Easy stuff (math) 4/4

• Growth formula
• A K (1g)t
• Time to double t ln2/ln (1g)
• Implicit Differentiation
• Regard y as a function of x
• yx2 y2 x3
• f(x)x2 (f(x))2 x3 gt now derive
• f (x)x2 f(x)2x 2(f(x))f (x) 3x2
• yx2 y2x 2yy 3x2

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Easy stuff (stats) - Intuition 1/4

• OLS Estimators
• Based on a sample, estimators for the coefficient
  of the population are calculated.
• Unbiasedness of the OLS Estimator
• Needs to be established to properly use these
  estimators on average we hit the true value
• 4 Assumptions SLR 1 4
• SLR1 Linear in parameters
• SLR2 Random sampling
• SLR3 Sample Variation (not all the same)
• SLR4 Zero Conditional Mean error unrelated to x

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Easy stuff (stats) - Intuition 2/4

• Omitted Variable Bias
• The bias in a model caused by omitting a variable
  with an effect. Example (thx. to Tobi)
• Take a random sample of 2 people
• They differ in basic ability, approximated by IQ
• Examine the relation of years of education to
  wage

IQ140
wage
IQ90
educ
10
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Easy stuff (stats) - Intuition 3/4

• F-test
• Used to test, whether a bunch (or all) of the
  coefficients are unimportant (ß0 ß1 ß2 0) or
  alternatively, to compare models.
• For the second purpose, one needs a restricted
  and one unrestricted model.
• The higher the F-score, the better for the
  validity of the model overall and also in favor
  of the unrestricted model.
• Distribution is non-quadratic and uses df
• Table in Bowerman

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Easy stuff (stats) - Intuition 4/4

• Indirect t-test (Teta-trick)
• Test for linear restriction
• Y ß0 ß1x1 ß2x2
• Define ? ß1 ß2
• Solve for ß1 and plug in
• Y ß0 (? - ß2)x1 ß2x2
• Shuffle and run regression
• Y ß0 ?x1 ß2(x2-x1)
• The t-score of the ?-coefficient can be used to
  evaluate joint significance

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Harder stuff 1 (QM3) 1/15

• Discrete Dynamics and Difference Equations
• General Idea Find a function that allows us to
  calculate the value of a series that changes over
  time at a specific point in time, where the
  periods of time are discrete
• ?K is the basis for this function!!
• e.g. Value of a depot after t years with g
  interest
• Growth formula relation

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Harder stuff 1 (QM3) 2/15

• General Notation
• Index-notation Kt, Kt-1
• Domain N 0,1,2,3, ...
• Kt Kt-11.05 Kt-2 (starts getting useful
  after the second period)
• Easy to do in Excel
• However, the example equation does not provide an
  answer to the question posted earlier!

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Harder stuff 1 (QM3) 3/15

• Assume a depot is growing with 5 each year. How
  much money do you get after t years, if you put
  in 1000 once?
• Kt 1000 (1.05)t gt growth formula
• Assume that you put in 1000 each year ... ?
• Kt Kt-1 (1.05) 1000
• That is the rule, lets try to solve it

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Harder stuff 1 (QM3) 4/15

• Classification
• Autonomous system (t does not occur as a factor,
  just as an index)
• Order of a system the highest time lag
• e.g. Kt Kt-1 1.05 Kt-3 is 3rd order
• Homogeneous system Discrete, 1st order,
  Autonomous

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Harder stuff 1 (QM3) 5/15

• Coming back
• Kt Kt-1 (1.05) 1000
• This is not easy to solve, due to the 1000,
  thus we ignore them for the moment, searching for
  the solution of the homogeneous equation first.
• This is, as we know, Kt K0 (1.05)t, with K0
  being the initial amount saved
• The solution of homogeneous equations always
  works like this!

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Harder stuff 1 (QM3) 6/15

• Lets try the inhomogeneous equation (IHE)
• Kt Kt-1 (1.05) 1000
• Solution to the homogeneous equation
• Kt K0 (1.05)t
• For the IHE, we need a different starting point,
  so we choose c and add a particular solution
  (pt)
• Kt c (1.05)t pt
• The c will be determined from a given K0, but
  first we need to find a pt

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Harder stuff 1 (QM3) 7/15

• Remember the original equation
• Kt Kt-1 (1.05) 1000
• The pt will always be similar to the stuff that
  makes the equation inhomogeneous ... so a
  constant in this case. Lets choose d
• Furthermore, remember it is a solution, so it can
  be plugged in for Kt. Since a constant does not
  vary in time, Kt Kt-1 d
• d d (1.05) 1000

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Harder stuff 1 (QM3) 8/15

• Solving for d gives us 20.000 as pt
• Kt c (1.05)t pt
• Kt c (1.05)t 20.000
• Remarks
• If I had K0, I could plug in t 0 and calculate
  c!
• pt can be a constant, a linear or quadratic
  equation with t as a variable, but is always
  similar to the part that makes the equation
  inhomogeneous

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Harder stuff 1 (QM3) 9/15

• Lets reconsider
• We wanted to solve an IHE
• Solved homogeneous system first (easy)
• Exchanged K0 with c and added a pt
• Choose a pt, similar to the thing that makes the
  homogeneous equation inhomogeneous (here d)
• Calculate pt, have complete solution (with K0
  even unique one)
• gt 1st order linear Difference Equation

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Harder stuff 1 (QM3) 10/15

• Example 2
• Kt 0.25 Kt-1 150 with K0 100
• Solution to the homogeneous Equation
• Kt K0 (0.25)t
• Substitute c and add pt
• Kt c (0.25)t pt
• pt like a constant, choose d for original
• d 0.25 d 150
• Solve for d and plug in
• d 200 gt Kt c (0.25)t 200
• Use K0
• 100 c (0.25)0 200 gt c 100
• Kt 100 (0.25)t 200

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Harder stuff 1 (QM3) 11/15

• Example 3
• Kt 1.5 Kt-1 10t 5
• Solution to the homogeneous equation
• Kt K0 (1.5)t
• Substitude c and add pt
• Kt c (1.5)t pt
• pt is like a linear equation, say at b
• at b 1.5 (a ( t-1) b) 10t 5
• at b (1.5a 10)t 5 1.5a 1.5b
• a 1.5a 10 and b 5 1.5a 1.5b
• a 20 , b 70
• Kt c (1.5)t 20t 70
• No K0 given

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Harder stuff 1 (QM3) 12/15

• Sidestep for equilibria
• An equilibrium point is a constant x, which will
  fulfill the original equation
• Kt Kt-1 (1.05) 1000
• Hence in this case x x (1.05) 1000, which
  turned out to be 20000
• An equilibrium is stable, if the difference
  equation moves towards this point, given any
  initial K0
• Kt c (1.05)t 20.000 gt not stable
• Kt c (0.05)t 20.000 gt -20.000

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Harder stuff 1 (QM3) 13/15

• 2nd order linear Difference Equations
• xt cxt-1 dxt-2
• Solving via Characteristic Equation
• ?2 c? d
• Solve for ? via pq-formula or ABC
• If you get two ?, the solution is
• xt k1?1t k2?2t
• If you get one ?, the solution is
• xt k1?t k2t?t
• If x0 and x1 are given, one can easiliy
• calculate k1 and k2

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Harder stuff 1 (QM3) 14/15

• If the original equation is an IHE
• xt cxt-1 dxt-2 bt
• Solve the homogeneous equation as usual
• Search for pt choose something similar to bt,
  plug it into the original equation for xt,
  xt-1,and xt-2
• The complete solution (second order linear
  difference equation) is the solution for the
  homogeneous equation pt (or further calculated
  with the help of x0 and x1)

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Harder stuff 1 (QM3) 15/15

• From the Syllabus for Discrete Dynamics
• 1.3 a), b) equilibria
• 1.7 a), b), c)
• 1.9 a), c)
• 1.12

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Harder stuff 2 (QM3) 1/7

• Continuous Dynamics and Differential Equations
• Similar to discrete dynamics, but now t is
  continuous.
• Again the basis of our analysis will be the
  change in K or x over time, denoted as and

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Harder stuff 2 (QM3) 2/7

• Lets try to solve this
• axt
• This means we are searching for a function xt,
  which is itself multiplied by a after deriving
  it.
• The e-function comes to mind
• If xt Aeat, then aAeat axt
• Remember, that is a derivative w.r.t. t!!

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Harder stuff 2 (QM3) 3/7

• Two more general rules for first order (linear)
  difference equations
• axt b gt xt Ae-at
• axt bxt2 gt xt
• Plus the third, which you already have seen
• axt gt xt Aeat
• Of course, knowing xo, will allow you to
  calculate A, since t 0 and provide a unique
  solution

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Harder stuff 2 (QM3) 4/7

• What you just have seen was a solution for a
  homogeneous equation.
• For IHE, the way to solve is the same as with
  discrete dynamics
• Solve the homogeneous equation
• Add a pt (related to the stuff that causes the
  IHE)
• Choose a pt, plug it into the original and solve
  it
• With a given x0 you can deduce the A for a unique
  solution

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Harder stuff 2 (QM3) 5/7

• First order and second order differential
  equations can be distinguished as in the case of
  difference equations
• First order xt axt-1, axt
• Second order xt axt-1 axt-2, a bxt

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Harder stuff 2 (QM3) 6/7

• Trying to solve 2nd order differential equation
• This is very similar to difference equations
• Take the differential equation and make the
  characteristic equation
• a bxt gt ?2 a ? b
• Solve this for ? by pq or ABC
• With 2 ? xt Ae ?1t Be?2t
• With 1 ? xt Ae ?t Bte ?t
• A and B can be uniquely determined using an x0
  and x1

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Harder stuff 2 (QM3) 7/7

• For IHE, the way of solving is again the same
  with solving the homogeneous equation, adding a
  pt, solving this with the usual trial and error
  approach and later on work on finding A and B.

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Easier stuff QM3 Equilibria 1/6

• Lets refresh the equilibria for difference
  equations
• An equilibrium point is a constant x, which will
  fulfill the original equation
• It is stable, if no matter what x0 is you end up
  in this x if t? 8
• For differential equations, the rules are the
  same
• Remember that x is a constant, thus 0

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Easier stuff QM3 Equilibria 2/6

• Consider (x-1)(x-3)
• The resulting equilibria are x 1 and x 3
• Graphically, this can be shown like this

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Easier stuff QM3 Equilibria 3/6

• Conergences
• And 1-dimensional

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Easier stuff QM3 Equilibria 4/6

• For phase diagrams, just remember
• Draw the graph of 0. The areas where the
  value of that function are gt 0, go right, else go
  left
• If two arrows meet in an equilibrium, it is
  stable, otherwise it is not

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Easier stuff QM3 Equilibria 5/6

• This also works for systems of differential
  equations with and , where both equations
  have to equal zero in order to have an
  equilibrium.
• Set the equations each equal to zero and draw
  them sequentially, each time drwaing your arrows.
  For right (gt0) and left (lt0) and for up (gt0)
  or down (lt0)
• Spiralling moves towards an intersection of your
  graphs will give you stable equilibria

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Easier stuff QM3 Equilibria 6/6

• Lets draw together
• x 1 and xy 2
• x 2y and x y

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Lecture 1

• by C. Kerkhoffs

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Announcement

• Thank you for your attention and please
• Come to our GMA tonight at 19.00 in A0.28

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