String Theory

MBA911 Quantitative Economics

Dr. Florian Gerth Faculty of Business [email protected]

What did we see in Session 1? • What is economics? • Production possibilities frontier and opportunity costs • Market demand and supply • Markets in action

• Market equilibrium • Surplus vs. shortage • Price ceilings vs. price flows

• Elasticities

• Probability concepts • Probability distributions

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What are we talking about in Session 2 • Measuring the size of the economy • Business Cycles (short run) • Economic Growth (long run) • Inflation • Unemployment

• Regression Analysis

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Watch videos (40 minutes):

• Gross Domestic Product and Economic Activity (14 minutes)

• Components of GDP (5 minutes)

• Business Cycles (11 minutes)

• Economic Growth (9 minutes)

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Chapter 11 – Measuring the size of the economy

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Gross domestic product (GDP) • GDP measures the market value of all final goods and services

produced within a nation’s geographic borders during a period of time

• It is a measure of a nation’s economic performance • There is an emphasis on GDP because it is the measure that is

most relevant to the level of domestic economic activity and generation of domestic jobs

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What does GDP include?

1. GDP counts only new domestic production • Excluding second-hand transactions

• Only change of ownership ≠ current production • Excluding ‘non-productive’ financial transactions

2. GDP counts only final goods • Excluding intermediate products • To prevent double-counting

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Concept of Value Added • GDP also can be used to measure how much value each stage

of production adds to the overall price of the final good or service

• The following slide shows how a $420 sale price of a woollen suit is split up over its stages of production

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Concept of Value Added $ value added at each supply chain Factor cost of suit Cumulative total Cost to …

$ value added by farmer $ 40.00 $ 40.00 Cost to supplier

$ value added by fabric supplier $ 110.00 $ 150.00 Cost to manufacturer

$ value added by suit manufacturer $ 150.00 $ 300.00 Cost to retailer

$ value added by retailer $ 100.00 $ 400.00 Factor price (excludes tax)

GDP at factor price $ 400.00 $ 420.00 Market price (includes tax)

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Measuring GDP • GDP counts the value of production in all markets for

products, resources, consumers, workers and businesses

• The circular flow model, shown in Exhibit 11.1 on the next slide, shows how we can think of all the economy’s sectors in one ‘whole’

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The circular flow model

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The circular flow model • The upper half of the diagram represents product markets

• HHs exchange money for goods and services they want and which are produced by forms

• Supply: from firms • Demand: from HHs

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The circular flow model • The lower half of the diagram represents factor markets

• Firms demand materials, land, labour, capital and entrepreneurship -factors or production- used to produce the goods and services sold in product markets

• Supply: from HHs • Demand: from firms  money payments (wages, interest) in

return

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Flow versus stock

• Flow is a rate of change in a quantity during a given time period; e.g.: • weekly consumption spending

• Stock is a quantity measured at one point in time; e.g.: • the amount of money in a bank account

• All measurements in the circular flow model are flows

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A four-sector circular flow model

• HHs • Firms • Financial Markets • Government • Foreign Markets

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A four-sector circular flow model • Most household income is spent on domestic goods • However:

• Part of household income is saved • Part of household income is taxed • Part is spent on goods from overseas (imports)

• These parts are called leakages from the main flow

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A four-sector circular flow model

• Spending also enters the circular flow from any of the following:

• Firms investing in new plants and equipment • Government spending on consumption and capital items • Foreigners purchasing exports

• These parts are called injections into the main flow

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The circular flow model of an open economy

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Approaches to measuring GDP

• Expenditure approach: Adds all spending for final goods during a period of time

• Income approach: Adds the incomes of all factors of production

• Both should yield the same result!

GDP = C + I + G + (X – M)

GDP = Wages + Rent + Interest + Profits

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The expenditure approach

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Household consumption (C) expenditures • These are the largest component of GDP (≈2/3) • They are made up of spending on:

• services (e.g. education, medical visits, haircuts, etc.) • durable goods (e.g. cars, appliances, furniture) • non-durable goods (e.g. food, clothing, petrol).

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Gross private domestic investment (I) • Investment = Capital formulation • This is spending that maintains or increases the stock of

capital equipment in the economy • It is made up of:

• fixed investment for newly produced capital goods • changes in business inventories (unsold finished goods and raw

materials)

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Gross private domestic investment (I) • Investment refers to spending on physical products that are

intended to be used as inputs in the production process

• Capital (or capital stock) refers to existing stock of buildings, factories and machinery

• Note: Investment is a flow, while capital is a stock

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Government consumption and investment expenditures (G)

• This includes: • the value of all goods and services that the government purchases • spending on investment-type goods, such as highways, buildings

and bridges

• G does not include transfer payments from one level of government to another.

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Net Exports (X – M)

• Exports (X) are expenditures by foreigners on domestically produced goods and services

• Imports (M) are the dollar amount of a nation’s purchases from producers in other countries

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GDP in other countries

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Nominal GDP (2018)

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What do you think about this figure? Is it biased or representative for the wealth of a nation?

Nominal GDP per capita (2018)

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What do you think about this figure? Is it biased or representative for the wealth of a nation?

Nominal GDP growth (2018)

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GDP’s shortcomings as measure of economic welfare

• GDP is seen as not being able to fully measure issues of economic welfare; e.g.:

• non-market transactions • distribution, kind and quality of products • neglect of leisure time • the underground economy • ‘economic bads’

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Nominal GDP and real GDP

• Nominal GDP is the value of all final goods and services produced during a given time period based on the prices existing during the time period of production

• Nominal GDP is also referred to as current-dollar GDP. • Real GDP is the value of all final goods and services produced

during a given time period based on the prices existing in a selected reference year

• Real GDP is also referred to as constant-dollar GDP

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Changing nominal GDP to real GDP • To get a clearer picture of an economy’s growth, it is necessary to

adjust nominal GDP to real GDP so that it reflects only changes in output and not changes in prices.

• Nominal GDP is converted to real GDP using the following conversion equation:

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Nominal GDP and real GDP

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Relationship between national saving and investment

• In a simple economy, saving is by definition its investment (S = I) • National saving = Total income – Consumption • National investment = Total spending on capital goods by private

sector and government • If national investment exceeds national saving, the additional

investment must be financed by net foreign investment flowing into the country

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Chapter 12 Business Cycle and Economic Growth

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The business cycle ‘roller-coaster’

• The business cycle alternates between periods of economic growth and contraction, which can be dated by changes in output, income, sales and employment measures

• Business cycles are inherent to market economies

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Two phases of the business cycle

Expansion • An upturn in the business cycle

during which measures of aggregate economic activity rise

• Phase of the business cycle between the trough and the peak

Recession • When output, sales and employment

decline in the economy

• Phase of the business cycle between the peak and the trough

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Hypothetical and actual business cycle

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Australia’s Economic Growth Trend

• Economic growth is an expansion in national output measured by the annual percentage increase in a nation’s real GDP

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Real GDP growth rates in selected countries

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Business cycle indicators • A number of macroeconomic variables provide information

regarding where the country is on the business cycle ‘roller- coaster’.

• These variables are often grouped into: • leading indicators • coincident indicators • lagging indicators.

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Leading indicators • Leading indicators are variables that change direction before the

economy shifts from one business cycle phase to another; e.g.:

– average amount of work done – new businesses formed – new building permits – material prices – stock prices

– money supply – gross operating surplus of

companies – ratio of GDP deflator to unit

labour costs

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Coincident indicators • Coincident indicators are variables that change at about the same

time that the economy shifts from one business cycle phase to another; e.g.:

– unemployment rate – total civilian employment – household income

– gross domestic non-farm product.

– industrial production – retail sales.

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Total spending and the business cycle

• Changes in total or aggregate expenditure – also referred to as aggregate demand – are the principal cause of business-cycle variations in GDP

• GDP = C + I + G + (X – M)

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The GDP gap • The GDP gap is the difference between full employment real

GDP and actual real GDP in a particular period • Full employment does not literally mean an unemployment

rate of zero

GDP gap = potential real GDP – actual real GDP

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Full Employment Determined by: • Fictional unemployment • Seasonal unemployment • Structural unemployment

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Economic growth in the longer term

• Knowledge of the determinants of economic growth aids governments to develop better policies

• Economic growth results from growth in demand (capacity to buy goods and services) and growth in supply (capacity to produce)

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Solow model of economic growth

• The Solow model of economic growth sought to explain how consumption, saving, capital, labour and technological change combine in the longer term to determine a nation’s economic growth

• In the Solow model, technological change is assumed to be exogenous

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The determinants of growth • Output growth is the result of increases over time in:

• land • labour • Capital

• 𝑌𝑌 = 𝐴𝐴. 𝐿𝐿𝛼𝛼 .𝐾𝐾𝛽𝛽 .𝑅𝑅𝛾𝛾

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The determinants of growth • Output per unit of labour (real GDP) can only increase once the

quantity of capital per unit of labour is increased

• Household savings determine the level of investment possible

• Only way for long-lasting economic growth lies in Total Factor Productivity (TFP)  the way inputs are transformed into outputs

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Stylised aggregate production functions

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Stylised aggregate production functions

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Production factor accumulation = Investment • This depends on the amount of investment per person a country is

making • A nation’s output is either consumed or saved/ used for capital

accumulation • A nation’s output not consumed is the savings, which, in turn,

allow for investment • Higher levels of saving per person imply higher levels of

investment per person

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What to do now (125 minutes):

• MCQs MCQs #1 (30 minutes) • Tutorial Questions (30 minutes) • Tutorial Questions Discussion (20 minutes) • Discussion “Poland attacks EU aid as smoke and mirrors” (30 minutes) • Discussion (15 minutes)

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Watch videos (12 minutes):

• GDP, Unemployment and Inflation (6 minutes)

• Types of Unemployment (4 minutes)

• Inflation in Venezuela (2 minutes)

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Chapter 13 – Inflation and unemployment

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Meaning of inflation

• Inflation is an increase in the general (average) price level of goods and services in the economy

• Other associated terms: • Deflation: A decrease in the general (average) price level of goods and

services in the economy • Disinflation: A reduction in the rate of inflation

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Consumer price index (CPI)

• The consumer price index (CPI) is an index that measures changes in the average prices of consumer goods and services

• The CPI is sometimes called the cost-of-living index

• It includes only consumer goods and services

• It excludes items purchased by businesses and government

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Consumer price index (CPI) • The Australian Bureau of Statistics (ABS) conducts quarterly

surveys of the prices of items typically purchased by an average family (called the ‘market basket’)

• It is a fixed-weight price index – the composition of the ‘basket’ remains unchanged from one period to the next

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Composition of the Australian CPI

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How the CPI is computed

• To calculate the CPI, we determine the cost of the same ‘market basket’, but valued at current-year prices, using the following formula:

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How the CPI is computed

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Annual rate of inflation (∏)

• The annual rate of inflation is the percentage change in the CPI from one year to the next

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Australia’s Inflation Rate, 1950-2017

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World Inflation (2020)

Criticism of the CPI • The CPI is not a perfect measure of inflation for the following

reasons: • The use of one ‘typical’ or ‘average’ basket • The difficulty of adjusting for changes in quality • The impact of the substitution effect

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Demand-pull inflation • Demand-pull inflation is a rise in the general price level

resulting from an excess of total spending (demand) over supply

• Prices are pulled up by the pressure from buyers’ total expenditures

• Demand-pull inflation tends to occur when the economy is operating near capacity, as buyers try to outbid one another for available resources

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Cost-push inflation • Cost-push inflation is a rise in the general price level

resulting from an increase in the cost of production, irrespective of demand conditions

• Upward pressure on prices could be caused by cost increases for labour, raw materials, construction, equipment and interest loans

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Consequences of inflation • Shrinks income • Affects real interest rate (turns lenders into losers) • Affects investment and business decisions

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Shrinks income • Nominal (or money) income does not measure purchasing

power • Real income measures the amount of goods and services that

can be bought with nominal income:

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Affects real interest rate • The real interest rate is the nominal rate of interest minus the

inflation rate:

• Inflation lowers the real purchasing power of money • If the real interest rate is negative (i.e. inflation exceeds

nominal interest rate), lenders and savers lose at the expense of borrowers

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Affects investment and business decisions • A low and stable rate of inflation is conducive to efficient

decision-making • Economic agents can accurately forecast general price

increases • High inflation may cause over-investment in some areas (e.g.

assets and speculation) and under-investment in others (e.g. real productive capacity)

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Inflation on a rampage • Hyperinflation is an extremely rapid rise in the general price

level. It often leads to rapid political and social change: • It encourages immediate spending • It jeopardises debtor–lender contracts • It sets in place a vicious wage-price spiral • It encourages investment on non-productive things

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Meaning of employment • In Australia, persons over 15 years of age working at least

one hour per week in paid employment or working at least 15 hours per week of unpaid work in a family business are classified as employed

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Measurement of unemployment • The unemployment rate is the percentage of people in the

labour force who are without jobs and are actively seeking jobs

• The civilian labour force comprises those people aged 15 years and older who are employed or who are actively seeking a job, excluding those in the armed forces, homemakers, students, discouraged workers and other persons not in the labour force

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Population, Employment and unemployment

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Australia’s Unemployment Rate, 1978-2017

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Unemployment Rates for Greece

7.70 9.50

12.50

17.70

24.20 22.10

25.80

32.90

44.40

55.30

0.00

10.00

20.00

30.00

40.00

50.00

60.00

2008 2009 2010 2011 2012

Unemp female

Unemp male

Unemp total

Unemp15 - 24

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What happened in 2008?

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Unemployment in other countries

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Criticism of the unemployment rate • Unemployment data might overstate true unemployment if

respondents in the ABS survey falsely report that they are seeking employment

• The rate could understate the true rate: • Discouraged workers are not counted • Underemployment: Many people would like to work longer hours in

different roles

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Types of unemployment • There are four conceptual categories of unemployment:

• Understanding these enables appropriate policies to be developed

Seasonal Frictional Structural Cyclical

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Seasonal unemployment • Seasonal unemployment is caused by recurring changes in

hiring due to changes in seasonal demand or weather conditions; e.g.:

• summer-resort workers • the retail trade during the Christmas period • construction workers in the wetter seasons

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Frictional unemployment • Frictional unemployment is caused by the normal search time

required by workers with marketable skills who are changing jobs, entering or re-entering the labour force

• It is sometimes called search unemployment • It is usually short-term and is a normal condition in an

economy that permits freedom of choice

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Structural unemployment

• Structural unemployment is caused by a mismatch between the skills of workers who are out of work and the skills required for existing job opportunities

• It results from changes in the structure of the economy over time (e.g. tastes, patterns of demand, technology)

• It is often long-term and may necessitate worker retraining

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Structural unemployment • Causes of structural unemployment include:

• inappropriate education or job-related skills • changes in tastes and demand over time • changes in production technology • competition from other markets • geographic differences.

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Cyclical unemployment • Cyclical unemployment is attributable to the business cycle

and is caused by the lack of sufficient jobs during a recession • It may cause unemployment rates to rise rapidly, but they are

often slow to fall again • Cyclical unemployment is a focus of macroeconomic policy

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The goal of full employment

• Full employment does not mean ‘zero per cent unemployment’; a certain level of frictional and seasonal unemployment is normal

• Full employment therefore defines the rate of unemployment that exists without cyclical unemployment

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Hysteresis • Hysteresis occurs when the full employment rate of

unemployment increases (decreases) as the actual unemployment rate increases (decreases)

• This can be caused by: • reduction in skills due to long periods of unemployment • the insider/outsider phenomenon

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Consequences of unemployment • The monetary cost of unemployment is the GDP gap – the

loss of potential output that could have been produced by the unemployed.

• Non-monetary costs include: • loss of self-worth • the unequal burden on labour-market groups within the labour

force (e.g. young people)

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Is there a relationship between inflation and unemployment?

• Exhibit 13.11 shows the relationship between unemployment and inflation, as illustrated by the short-run and long-run Phillips curves

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Is there a relationship between inflation and unemployment?

• AW Phillips: Argues that an inverse relationship exists based on demand-pull inflation

• M Friedman: Argues that such trade-offs would only be very short- run in nature, and that any attempt to lower the unemployment rate below the ‘non-accelerating inflation rate of unemployment’ would eventually raise the rate of inflation, thereby increasing inflationary expectations, which further increase the actual inflation rate

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What to do now (80 minutes):

• MCQs MCQs #2 (30 minutes) • Tutorial Questions (30 minutes) • Tutorial Questions Discussion (20 minutes)

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Watch videos (18 minutes):

• Aggregate Demand (7 minutes)

• Aggregate Supply (7 minutes)

• AS-AD (4 minutes)

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Chapter 14 – A simple model of the macroeconomy

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Classical school of economics

• Adam Smith: Founder of the classical school of economics

• Classical economists believed that: • supply and demand would achieve full employment, because flexible

prices in competitive markets bring all markets into equilibrium • markets are always clear, enabling firms to sell all goods and services

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Introducing the Keynesian revolution

• John Maynard Keynes observed the high rates of unemployment in the Great Depression, which persisted for many years

• In The General Theory of Employment, Interest and Money (1936), he challenged classical thinking by placing great primacy on the demand side of the economy rather than the supply side

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Economic determinants of the four key expenditure components

Aggregate demand is the sum of:

• consumption demand • investment demand • government demand • net export demand

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Consumption demand (C) • This is the largest component of aggregate expenditure

• The most important determinant of consumption demand is disposable income

• This relationship is represented by the consumption function (C): the amount that households want to spend on goods and services at different levels of disposable income.

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Marginal propensities to consume and save • The marginal propensity to consume (MPC) is the change in

consumption resulting from a given change in real disposable income

• The marginal propensity to save (MPS) is the change in savings resulting from a given change in real disposable income

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Factors other than income affecting consumption • Expectations: Optimistic or pessimistic views about the future can

change C patterns today

• Wealth held as real and financial assets

• The price level

• The interest rate: Lower rates encourage borrowing to finance C, and vice versa

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Investment demand (I) • This is the most volatile component of aggregate expenditure,

since it is future-oriented and risky

• It is based on: • firms’ expectations (i.e. future returns) • interest rates; ceteris paribus, low rates reduce the risks of

investment and the cost of borrowing

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Other factors affecting Investment demand

• Technological progress made by firms • Capacity utilisation by firms • Government policies, such as taxation

• 𝐼𝐼 = 𝑓𝑓(𝑖𝑖(−),𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝐸𝐸𝐸𝐸𝐸𝐸(+))

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Government demand (G)

• Government expenditure is autonomous. • It does not vary according to the current level of disposable

income or interest rates. • This means that government spending is primarily the result of

a political decision made independently of the level of national output or interest rates.

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Net Export Demand (X-M) • Economic conditions overseas affect the demand for a

country’s exports, and domestic conditions affect the demand for imports

• Exchange rates and the terms of trade (relative price movements of exports and imports) also affect net export demand and its dollar value

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The Aggregate Demand-Output Model

• The aggregate demand–output model equates the output produced to the level of aggregate demand

• Sum of the four categories (C+I+G+X-M)

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Aggregate demand • Buyers – including households, government, other firms and

foreigners – are willing to purchase the outputs of firms

• The sum of their ‘willingness to purchase’ goods and services is the aggregate demand (AD):

AD = C + I + G + (X – M)

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The spending multiplier effect

• This refers to induced rounds of spending that occur in an economy after some initial stimulus to spending

• Any change to the level of aggregate demand in the Keynesian model starts a chain of other changes  consumption function

• This ripple effect results in an amplified change to the level of real GDP

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The Aggregate Demand-Aggregate Supply Model

• To be used in modern business cycles, the simple, demand- oriented Keynesian framework must be augmented by placing some emphasis on aggregate supply

• That is, the aggregate demand in an economy is linked to the economy’s aggregate supply.

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The aggregate demand curve

• The curve (shown on the next slide) shows the level of total real GDP purchased by all participants in the economy at different price levels during a time period, ceteris paribus

• The curve is downward-sloping, suggesting that the lower the economy-wide price level, the greater the aggregate quantity demanded for goods and services

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The aggregate demand curve

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Reasons for the AD curve’s shape

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Non-price-level determinants of Aggregate demand

Non-price-level determinants

(C, I, G, X and M) shift the aggregate demand

curve.

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The aggregate supply curve

• The aggregate supply curve shows the level of real GDP that firms would be willing to produce at different price levels during a time period, ceteris paribus

• It is upward-sloping because firms will be induced to bring more products to the market if the price level rises

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The horizontal aggregate supply case

An increase in aggregate demand (AD) results in increased real

GDP. Prices and wages are ‘sticky’ because of idle resources.

Therefore, the aggregate supply curve is horizontal.

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The vertical aggregate supply case

An increase in AD results in a price increase and an increase in

production. This raises the cost of production and price, which reduces

quantity of AD.

Therefore, the aggregate supply curve is vertical.

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Three ranges of the AS curve

• The Keynesian view is more likely to be true in a recession

• The classical view is more likely to hold at full employment output

• Between these two ranges is the intermediate range, when the aggregate supply curve rises as the economy approaches full employment.

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Three ranges of the AS curve

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Macroeconomic equilibrium • Macroeconomic equilibrium occurs when the aggregate demand and

aggregate supply curves intersect

• At this point, sellers neither overestimate nor underestimate the real GDP demanded at a particular price

• Any difference between aggregate demand and aggregate supply will bring about a change in firms’ inventories, and also cause firms to adjust their plans for the future.

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Macroeconomic equilibrium: AD-AS model

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Changes in the AD-AS macroeconomic equilibrium • The business cycle is explained as the aggregate demand curve

shifts along a stationary aggregate supply curve

• The effects of changing aggregate demand on real output and the price level depend upon the range of the aggregate supply curve

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Horizontal range

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Intermediate range

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Vertical range

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Equilibrium shifts arising from AS shifts

A shift of the aggregate supply curve can be

caused by a change in:

• resource prices • technology

• taxes • subsidies

• regulations.

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Cost-push and demand-pull inflation revisited

• Cost-push inflation: A rise in the price level due to an inward shift of the aggregate supply curve, while the aggregate demand curve remains unchanged

• Demand-pull inflation: A rise in the price level resulting from an inward shift in the aggregate demand curve, while the aggregate supply curve remains unchanged

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Shift factors of AD and AS

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What to do now (125 minutes):

• MCQs MCQs #3 (30 minutes) • Tutorial Questions (30 minutes) • Tutorial Questions Discussion (20 minutes) • Discussion “Zimbabwe pushed to bring of famine” (30 minutes) • Discussion (15 minutes)

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Watch videos (18 minutes):

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• Linear Regression I (5 minutes)

• Linear Regression II (13 minutes)

Install Analytic Toolpack in Excel: • Analytic Toolpack in Excel

Additional files for block #4

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• Excel file: “Excel-file – Regression Analysis – Block #4”

• Pdf file: “F-table (5%)”

Render - Chapter 5 Regression Analysis

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Regression Models/ Analysis

• Regression analysis – powerful tool! • 1st purpose: Understand the relationship between variables • 2nd purpose: Predict the value of one variable based on another variable  forecasting

• Simple linear regression models have only two variables  𝑦𝑦 = 𝛽𝛽0 + 𝛽𝛽1𝑋𝑋 + 𝜖𝜖

• Multiple regression models have more than one independent variable  𝑦𝑦 = 𝛽𝛽0 + 𝛽𝛽1𝑋𝑋 + 𝛽𝛽2𝑋𝑋 + ⋯+ 𝛽𝛽𝑖𝑖𝑋𝑋 + 𝜖𝜖

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Notes on board: What is an equation and how does a graph look like

Introduction • Variable to be predicted is called the dependent

variable (Y) or response variable • Value depends on the value of the independent variable(s) • Explanatory or predictor variable (X)

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Y X1 X2

Questions that can be answered with Regression Models

• What is the relationship between… • Advertising costs and sales • Income and consumption • Gross Domestic Product (GDP) and Investment • Hours of studying for ECON332 and final exam results • Age and price of a car • …

Can we obtain a mathematical rule (formula) between both occurrences, X and Y, which shows us the strength of this relationship?!

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Scatter Diagram/ Plot

• Scatter diagram or scatter plot is often used to investigate the relationship between the variables

• Independent variable normally plotted on X axis • Dependent variable normally plotted on Y axis

• It is a graph of the data that helps us to visualise and understand the relationship

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Triple A Construction • Triple A Construction renovates old homes • The dollar volume of renovation work is dependent on the area payroll

(averages wages for people living in this area)

Triple A Construction Company Sales and Local Payroll SALES

($100,000s) LOCAL PAYROLL

($100,000,000s)

6 3 8 4 9 6 5 4 4.5 2 9.5 5 Dr. Florian Gerth - MBA911 - Quantitative Economics

137

Triple A Construction

Scatter Diagram

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SALES PAYROLL

6 3 8 4 9 6 5 4 4.5 2 9.5 5

The plot indicates that higher values for the local payroll seem to result in higher sales for the company. There is not a perfect relationship because not all points lie in a straight line, so there would be some error involved if we tried to predict sales based on payroll using this or any other line.

Triple A Construction

Scatter Diagram

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What does this line tell us? Can we calculate (estimate) it? Why would that be useful?  This is what Regression Analysis is all about!

Many lines could be drawn through these points, but which one best represents the true relationship? Regression analysis provides the answer!

Simple Linear Regression • Regression models used to test relationships between variables

0 1Y Xβ β ε= + +

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where Y = dependent variable (response) X = independent variable (predictor or explanatory) β0 = intercept (value of Y when X = 0) β1 = slope of the regression line e = random error

This is the TRUE relationship. We will, however, never be able to calculate that. Therefore, we estimate (approximate) it using sample data.

Simple Linear Regression • True values for the slope and intercept are not known

• Estimated using sample data

0 1Ŷ b b X= +

where Ŷ = predicted value of Y

b0 = estimate of β0, based on sample results b1 = estimate of β1, based on sample results

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Triple A Construction Predict sales based on area payroll

Y = Sales X = Area payroll

The line in the scatter plot (slide 138) minimises the errors

Error = (Actual value) − (Predicted value)

ˆe Y Y= −

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The best regression line will be defined as the one with the minimum sum of the squared error  Regression analysis  Least Squares Regression

Triple A Construction Formulas for simple linear regression, intercept and slope

0 1Ŷ b b X= +

1 2

0 1

average (mean) of values

average (mean) of values

( )( ) ( )

X X X

n Y

Y Y n

X X Y Y b

X X

b Y b X

= =

= =

− − =

−

= −

∑

∑

∑ ∑

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Triple A Construction

Regression Calculations

Y X (X − X̅)2 (X − X̅)(Y − Y)̅ 6 3 (3 − 4)2 = 1 (3 − 4)(6 − 7) = 1 8 4 (4 − 4)2 = 0 (4 − 4)(8 − 7) = 0 9 6 (6 − 4)2 = 4 (6 − 4)(9 − 7) = 4 5 4 (4 − 4)2 = 0 (4 − 4)(5 − 7) = 0 4.5 2 (2 − 4)2 = 4 (2 − 4)(4.5 − 7) = 5 9.5 5 (5 − 4)2 = 1 (5 − 4)(9.5 − 7) = 2.5

ΣY = 42

Y̅ = 42÷6 = 7

ΣX = 24

X̅ = 24÷6 = 4

Σ(X − X)̅2 = 10 Σ(X − X)̅(Y − Y̅) = 12.5

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Triple A Construction

Regression calculations

24 4 6 6 X

X = = =∑ 42 7 6 6 Y

Y = = =∑

1 2

0 1

( )( ) 12.5 1.25 10( )

7 – (1.25)(4) 2

X – X Y –Y b

X – X

b Y b X

= = =

= − = =

∑ ∑

Therefore ˆ 2 +1.25Y X=

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Triple A Construction

Regression calculations

24 4 6 6 X

X = = =∑ 42 7 6 6 Y

Y = = =∑

1 2

0 1

( )( ) 12.5 1.25 10( )

7 – (1.25)(4) 2

X – X Y –Y b

X – X

b Y b X

= = =

= − = =

∑ ∑

Therefore ˆ 2 +1.25Y X=

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sales = 2 + 1.25(payroll)

If the payroll next year is $600 million

�𝒀𝒀 = 2 + 1.25(6) = 9.5 or $ 950,000

Notes on board

This model tells us that each time the payroll increases by 1 (represented by X), we would expect the sales to increase by 1.25.

Measuring the Fit of the Regression Model • Regression models can be developed for any variables X and Y

• How do we know whether the model is actually helpful in predicting Y based on X?

• Three measures of variability • SST – Total variability about the mean • SSE – Variability about the regression line • SSR – Total variability that is explained by the model

• SS = Sum of Squares Why is that? Dr. Florian Gerth - MBA911 - Quantitative Economics 147

Measuring the Fit of the Regression Model

• Sum of squares total

• Sum of squares error

• Sum of squares regression

• An important relationship

2SST ( )Y Y= −∑

2 2ˆSSE ( )e Y Y= = −∑ ∑

2ˆSSR ( )Y Y= −∑

SST SSR + SSE=

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Notes on board

Measuring the Fit of the Regression Model

Sum of Squares for Triple A Construction Y X (Y − Y̅)2 Ŷ (Y − Ŷ)2 (Ŷ − Y̅)2

6 3 (6 − 7)2 = 1 2 + 1.25(3) = 5.75 0.0625 1.563

8 4 (8 − 7)2 = 1 2 + 1.25(4) = 7.00 1 0

9 6 (9 − 7)2 = 4 2 + 1.25(6) = 9.50 0.25 6.25

5 4 (5 − 7)2 = 4 2 + 1.25(4) = 7.00 4 0

4.5 2 (4.5 − 7)2 = 6.25 2 + 1.25(2) = 4.50 0 6.25

9.5 5 (9.5 − 7)2 = 6.25 2 + 1.25(5) = 8.25 1.5625 1.563

Y̅ = 7 Blank ∑(Y − Y̅)2 = 22.5 Blank ∑(Y − Ŷ)2 = 6.875 ∑(Ŷ − Y̅)2 = 15.625

Blank Blank SST = 22.5 Blank SSE = 6.875 SSR = 15.625

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Measuring the Fit of the Regression Model

For Triple A Construction SST = 22.5 SSE = 6.875 SSR = 15.625

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Measuring the Fit of the Regression Model

Deviations from the Regression Line and from the Mean

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Coefficient of Determination • The proportion of the variability in Y explained by the regression equation

• The coefficient of determination is r2.

2 SSR SSE1– SST SST

r = =

• For Triple A Construction

2 15.625 0.6944 22.5

r = =

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Notes on board

Coefficient of Determination

About 69% of the variability in Y is explained by the equation based on payroll (X)

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Correlation Coefficient Degree or strength of the linear relationship

• Always between +1 and −1 • The correlation coefficient is r

2r r= ±

• For Triple A Construction

0.6944 0.8333r = =

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154

It is negative if the slope is negative, and it is positive if the slope is positive.

Four Values of the Correlation Coefficient Values of the Correlation Coefficient

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Assumptions of the Regression Model • With certain assumptions about the errors, statistical tests can be

performed to determine the model’s usefulness:

1. Errors are independent 2. Errors are normally distributed 3. Errors have a mean of zero 4. Errors have a constant variance

• A plot of the residuals (errors) often highlights violations of assumptions

• When the errors (residuals) are plotted against the independent variable, the pattern should appear random.

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Residual Plots

Pattern of Errors Indicating Randomness

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Residual Plots

Nonconstant Error Variance

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In general, patterns in the plot of the errors indicate problems with the assumptions or the model specification.

Residual Plots

Pattern of Errors Indicating Relationship Is Not Linear

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In general, patterns in the plot of the errors indicate problems with the assumptions or the model specification.

 Some other form of model, e.g., quadratic should be used.

Estimating the Variance

Errors are assumed to have a constant variance (σ2), usually unknown • Estimated using the mean squared error (MSE), s2

2 SSEMSE 1

s n k

= = − −

where n = number of observations in the sample k = number of independent variables

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𝜎𝜎2(not known)  𝐸𝐸2 (estimated through sample results)

Estimating the Variance

For Triple A Construction

2 SSE 6.8750 6.8750MSE 1.7188 1 6 1 1 4

s n k

= = = = = − − − −

• Estimate the standard deviation, s • The standard error of the estimate or the standard

deviation of the regression

MSE 1.7188 1.31s = = =

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Used in statistical tests about the model

Testing the Model for Significance

When the sample size is too small, you can get good values for MSE and r2 even if there is no relationship between the variables

• Testing the model for significance helps determine if the values are meaningful • Performing a statistical hypothesis test

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Testing the Model for Significance • We start with the general linear model

0 1Y Xβ β ε= + +

• If β1 = 0, the null hypothesis is that there is no relationship between X and Y • The alternate hypothesis is that there is a linear relationship (β1 ≠ 0) • If the null hypothesis can be rejected, we have proven there is a relationship • We use the F statistic

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Notes on board

Testing the Model for Significance – F-test

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Notes on board

SSRMSR k

= MSR MSE

F =

k = number of independent variables in the model

Describes an F distribution with: degrees of freedom for the numerator = df1 = k degrees of freedom for the denominator = df2 = n − k − 1

• If there is very little error, MSE would be small and the F statistic would be large – model is useful

• If the F statistic is large, the significance level (p-value) will be low, – unlikely would have occurred by chance

• When the F value is large, we can reject the null hypothesis and accept that there is a linear relationship between X and Y and the values of the MSE and r2 are meaningful

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Testing the Model for Significance – F-test

Notes on board

Steps in a Hypothesis Test

1. Specify null and alternative hypotheses

0 1

1 1

: 0 : 0

H H

β β

=

≠

2. Select the level of significance (α)  0.05 or 5%

3. Calculate the value of the test statistic

MSR MSE

F =

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Steps in a Hypothesis Test 4. Make a decision using one of the following methods

a) Reject the null hypothesis if the test statistic is greater than the F value from the table Otherwise, do not reject the null hypothesis:

1 2, ,

1

2

Reject if

1

calculated df dfF F

k n k

α>

=

= − −

df df

b) Reject the null hypothesis if the observed significance level, or p-value, is less than the level of significance (α). Otherwise, do not reject the null hypothesis:

-value ( calculated test statistic) Reject if -value p P F

p α = >

< Dr. Florian Gerth - MBA911 - Quantitative Economics

167

Triple A Construction Step 1

H0: β1 = 0(no linear relationship between X and Y) H1: β1 ≠ 0 (linear relationship exists between X and

Y)

Step 2 Select α = 0.05

Step 3 Calculate the value of the test statistic

SSR 15.6250MSR 15.6250 1

MSR 15.6250 9.09 MSE 1.7188

k

F

= = =

= = =

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Triple A Construction Step 4

Reject the null hypothesis if the test statistic is greater than the F value from the table

df1 = k = 1 df2 = n − k − 1 = 6 − 1 − 1 = 4

The value of F associated with a 5% level of significance and with degrees of freedom 1 and 4 is found the table.

F0.05,1,4 = 7.71 Fcalculated = 9.09 Reject H0 because 9.09 > 7.71

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Which table are you talking about?

The F table – 5%

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Triple A Construction F Distribution for Triple A Construction Test for Significance

• We can conclude there is a statistically significant relationship between X and Y

• The r2 value of 0.69 means about 69% of the variability in sales (Y) is explained by local payroll (X)

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Example

Advertising costs and Sales SALES

($100,000s) Advertising Costs

($100,000,000s)

34 3.4 22 2.5 54 5.7

43.5 4.6 51 5.4 37 3.5

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See Excel file

Questions?

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Remember! • Next lecture: 19th and 20th of November

• Reflective blog #3: due on 13th of November

• Economics in the Media Presentation: • Topic due 4th of November • Presentation due on 18th of November

• Research Project: • Topic due date: 20th of November • Due on: 2nd of December

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