Björn Andrén 🐻
    📉

    AI2152 - Quantitative Methods Applied to Real Estate

    Av: Björn Andrén & Ulf Nilsson

    Formler som använts under tidigare tentor

    Annuity

    Present Value Annuity Cashflows
    Present Value Annuity Cashflows

    Variance

    image

    Present Value

    Present Value
    Present Value

    Future Value of Cash today

    Cash (C0), discount rate, n = time
    Cash (C0), discount rate, n = time

    IRR (Internal Rate of Return)

    OBS! In excel CF0 needs to be negative
    OBS! In excel CF0 needs to be negative

    Price Elasticity (Midpoint method)

    The result indicates a 1% change in price will change the quantity by x %.
    image

    Multipel R, R-Square, Adjusted R-Square

    image

    Quantity Theory of Money

    image
    image

    Tenta 2021 October

    Question 1. Calculate the present value of Annuity cashflows and a Bonus payment

    Your investment pays you annual cash flows of $25 000 for the next 40 years. Year 40, your investment also pays you $500 000 (in addition to the $25 000). As usual all payments occur at the end of each year. Your required rate of return (the discount rate) is 5.5%. The present value of the cash flow is ___

    Combine two formulas

    Annuity

    image

    Present Value of future payment

    image
    Given
    Acronym
    Annual Cashflow
    25 000
    CF
    Bonus Payment Year 40
    500 000
    CFt
    Time Period Years
    40
    T
    Rate of Return
    5,5%
    r
    🧠
    Present Value = (Annuity Cashflows) + (Present Value of Bonus payment) → Present Value = CF*1/r*(1-(1/((1+r)^T))) + CFT/((1+r)^T)
    ‣
    Answer ✅

    A) $401 153 B) $2 500 000 → Not viable, money loses value over time C) $459 885 D) $974 174 → Not viable, money loses value over time E) None of the above (A, B, C, D) is close to be correct.

    Question 2 - Calculate the constant cashflow growth in perpetuity

    You assume that your new investment will generate annual cash flow forever (in perpetuity). Next year (year one) your investment will pay you $5 million. Thereafter, you assume that the yearly payments grow with an incredible (according to you) constant annual growth rate in perpetuity. Your required rate of return (the discount rate) is 12%. The present value of the growing cash flow is $62 500 000. Then the constant annual growth rate must be_____.
    Given
    Acronym
    Year One CF1
    $5 000 000
    CF1
    Required Rate of Return (Discount Rate)
    12%
    r
    Present Value of the growing Cashflow
    $62 500 000
    PV
    Constant growth rate
    (?)%
    g

    Present Value of growing cashflow in perpetuity

    image

    (It’s the last part of the formula we need, since they want us to calculate g)

    🧠
    PV = CF1/(r-g) → g = -((CF1/PV)-r)
    ✅
    g = 4%
    ‣
    Answer

    A) 6 % B) 5 % C) 4 %. ✅ D) 3 %. E) None of the above (A, B, C, D) is close to be correct.

    Question 3 - Expected IRR, Risk analysis, Variance, Standard Deviation

    A property can be purchased for 12 000 000 today (USD). A real estate analyst who likes risk analysis is analyzing the expected IRR, and the risk measured as the standard deviation, of the real estate investment by projecting five different scenarios as follows: Severe recession: NOI will be 800 000 the first year, and then decrease 4.5 percent per year until year six. The property will sell for 9 000 000 in year six. The probability for this scenario is 10 percent. Moderate recession: NOI will be 800 000 the first year, and then decrease 2.5 percent per year until year six. The property will sell for 10 000 000 in year six. The probability for this scenario is 15 percent. Baseline forecast: NOI will be level 800 000 per year for the next six years. The property will sell for 13 000 000 in year six. The probability for this scenario is 40 percent. Moderate expansion: NOI will be 800 000 the first year, and then increase by 2.0 percent per year until year six. The property will sell for 14 000 000 in year six. The probability for this scenario is 30 percent. Strong boom expansion: NOI will be 800 000 the first year, and then increase 4.0 percent per year until year six. The property will sell for 15 000 000 in year six. The probability for this scenario is 5 percent.

    Given from the question

    Scenario Prob. (%)
    Growth Rate (r)
    Salvage Value (Sale Year 6)
    Severe recession
    10%
    -4,5%
    9 000 000
    Moderate recession
    15%
    -2,5%
    10 000 000
    Baseline forecast
    40%
    0%
    13 000 000
    Moderate expansion
    30%
    +2%
    14 000 000
    Strong boom expansion
    5%
    +4%
    15 000 000
    Scenario Prob. (%)
    NOI Year Zero
    NOI Y1
    NOI Y2
    NOI Y3
    NOI Y4
    NOI Y5
    1️⃣ NOI Y6 (Sale + CF6)
    2️⃣ IRR Scenarios
    4️⃣ Scenario Variance
    Severe recession
    10%
    -12 000 000
    800 000
    764 000
    729 620
    696 787
    665 432
    9 635 487
    2,01%
    0,0266%
    Moderate recession
    15%
    -12 000 000
    800 000
    780 000
    764 400
    749 112,00
    734 130
    10 719 447
    3,80%
    0,0170%
    Baseline forecast
    40%
    -12 000 000
    800 000
    800 000
    800 000
    800 000
    800 000
    13 800 000
    7,81%
    0,0016%
    Moderate expansion
    30%
    -12 000 000
    800 000
    816 000
    832 320
    848966
    865 946
    14 883 265
    9,18%
    0,0121%
    Strong boom expansion
    5%
    -12 000 000
    800 000
    832 000
    865 280
    899891
    935 887
    15 973 322
    10,49%
    0,0055%
    3️⃣ Mean IRR (Expected IRR)
    7,173% ✅

    1️⃣ Future Value Cashflow

    To calculate NOI for year 1→6

    image

    2️⃣ Calculate IRR for all scenarios📗

    =IRR((NOI Year Zero:NOIY6))

    Obs → NOI Year zero måste vara negativt för att IRR formeln ska funka i excel

    3️⃣  Calculate Mean IRR (Expected IRR) 📗

    The sum of all scenarios based on their probability (Scenario probability)*(Case IRR)

    =SUMPRODUCT(Scenario Prob(1->5);IRR(1->5))

    4️⃣ Calculate Variance for each Case📗

    =Scenario Prob.*((IRR for case)-(expected IRR))^2

    5️⃣ Calculate Variance of Scenarios 📗

    =sum(all case Variances) --> Variance

    6️⃣ Standard Deviation 📗

    =SQRT(Variance)

    5️⃣ Variance
    0,0628%
    6️⃣ Standard Deviation
    2,506% ✅
    ✅
    Standard Deviation = 2,506%, Expected IRR = 7,173%
    ‣
    Answer

    A) The expected IRR is 7.17 % and the standard deviation is 2.52 %. ✅ B) The expected IRR is 7.17 % and the standard deviation is 0.06 %. C) The expected IRR is 9.19 % and the standard deviation is 5.82 %. D) The expected IRR is 0.5 % and the standard deviation is 0.06%. E) None of the above (A, B, C, D) is close to be correct.

    Question 4 - Expected NPV based on 5 Scenarios

    If the required rate of return (the discount rate) is 10 % for each of the five scenarios in question 10, then the expected NPV is____.
    Scenario Prob. (%)
    NOI Year Zero
    NOI Y1
    NOI Y2
    NOI Y3
    NOI Y4
    NOI Y5
    NOI Y6 Sale + CF6
    1️⃣ Net Present Value
    Severe recession
    10%
    -12 000 000
    800 000
    764 000
    729 620
    696 787
    665 432
    9 635 487
    -3 765 070
    Moderate recession
    15%
    -12 000 000
    800 000
    780 000
    764 400
    749 112,00
    734 130
    10 719 447
    -3 035 455
    Baseline forecast
    40%
    -12 000 000
    800 000
    800 000
    800 000
    800 000
    800 000
    13 800 000
    -1 177 630
    Moderate expansion
    30%
    -12 000 000
    800 000
    816 000
    832 320
    848966
    865 946
    14 883 265
    -454 258
    Strong boom expansion
    5%
    -12 000 000
    800 000
    832 000
    865 280
    899891
    935 887
    15 973 322
    277 248
    Given
    Acronym
    Required rate of Return (Discount Rate)
    10%
    i (r otherwise)
    Time Period
    6 Years
    t = 0→6
    NOI Year 0 → Year 6
    (See table above)
    Rt

    1️⃣ Calculate Net Present Value 📗

    image

    Net present Value = NOIY0 + (NOIY1/((1+r)^(1)) … (NOIY6/((1+r)^(6))

    2️⃣ Calculate Expected Mean NPV 📗

    For scenario analysis you get Expected Mean NPV based on the probability of each scenario.

    =SUMPRODUCT((Scenario Prob(1->5);NPV(1->5))

    =SUMPRODUCT((Prob1:Prob5;NPV1:NPV5)

    ✅
    Expected Mean NPV = -1 425 292 USD
    ‣
    Answer

    A) ‒1 177 630. B) 1 177 630. C) ‒1 428 788. ✅ (→Close enough) D) 1 428 788. E) None of the above (A, B, C, D) is close to be correct.

    Question 5 - Price Elasticity of rent price and apartment suppy

    Assume that an apartment rents for $950 per month and at that price the landlord rents out 12 000. When the price increases to $1 000 per month, the landlord supplies 15 000 units into the market. What is the correct interpretation of the price elasticity of supply? Use the midpoint method for elasticity.
    Before 1
    After 2
    Quantity
    12 000 (q1)
    15 000 (q2)
    Price
    950 (p1)
    1000 (p2)

    Formula Price Elasticity (Midpoint)

    OBS - The result indicates a 1% change in price will change the quantity by x %.
    OBS - The result indicates a 1% change in price will change the quantity by x %.
    ✅
    E = (0,22/0,05)% → E = 4,33%
    The result indicates a 1% change in price will change the quantity by x %.
    🧠
    The answers are tricky E = 4,33% It’s positive, hence a decrease in price (-1%) will result in a (-4,33%) decrease in supply.

    Answers

    A) A 1% decrease in the rent (price) will result in a 4.33% decrease in the quantity supplied. ✅ B) A 10% decrease in the rent (price) will result in a 4.33% decrease in the quantity supplied. C) A 4.33% rise in rent (price) will result in increase in quantity supplied of 1 %. D) A 4.33% rise in rent (price) will result in increase in quantity supplied of 10 %. E) None of the above (A, B, C, D) is close to be correct.

    Questions 6, 7 & 8

    Q7 - Maximum arithmetic mean quarterly return

    The HPI data shows quarterly international nominal house price index series (levels, not percentage returns). For each country compute the quarterly simple returns. The country that has the highest (maximum) arithmetic mean quarterly return is________.

    (Example of how to compute this in excel 📗 )

    Australia 🇦🇺
    Belgium 🇧🇪
    Canada 🇨🇦
    1975 Q1
    7.60
    15.18
    16.23
    1975 Q2
    7.74
    15.93
    16.46
    1975 Q3
    8.04
    16.74
    17.17
    1975 Q4
    8.29
    17.65
    17.41
    Calculated Q-Returns
    Australia 🇦🇺
    Belgium 🇧🇪
    Canada 🇨🇦
    1975 Q1
    (Q2-Q1)/(Q1)
    0.049
    0.013
    1975 Q2
    (Q3-Q2)/Q2
    0.05
    0.043
    1975 Q3
    0.031
    0.054
    0.014
    1975 Q4
    0.034
    0.057
    0.013
    Australia 🇦🇺
    Belgium 🇧🇪
    Canada 🇨🇦
    Mean
    =Average(Q1:Q4)
    =Average(🇧🇪  Q1:Q4)
    Max
    =Max(”Select Mean Row”)

    1️⃣ Calculate Quarterly Returns (Simple)📗

    (Q2 Returns - Q1) / Q1

    =(data Q(n+1)-data Q(n))/data Q(n)

    2️⃣  Compute Arithmetic Mean📗

    Arithmetic Mean = Average = Medelvärde

    Beräkna medelvärdet för vardera land i excell 🇨🇦 🇧🇪 🇦🇺

    =average(Select first Q: Last quarter data set)

    3️⃣ Find Max “Medelvärde” mellan länderna 📗

    =max(Select Cells mean Row)

    (Kolla vilket land som har Maxvärdet - går att ctrl-f:a excel 😊  )

    Resultat till vänster är ett metodexempel för hur man gör i Excell. För frågan har vi 25 länder och Quarterly returns från 1975→ 2021.
    ‣
    Answer

    A) Slovenia with arithmetic mean return of 9.30 %

    B) Israel with arithmetic mean return of 6.79 %

    C) Sweden with arithmetic mean return of 11.62 %

    D) Croatia with arithmetic mean return of 11.62 % ✅

    E) None of the above (A, B, C, D) is close to be correct.

    Q9 - Find Maximum and Minimum in a huge set of data

    The lowest (minimum) and highest (maximum) simple return observed among all countries and quarters is _________.

    1️⃣ Find Max and Min Value 📗

    =max(All Monthly return cells for all countries)

    =min(All Monthly return cells for all countries)

    🧠
    Big-brain Move → Select Cells with class 🎩  1. Välj första Cellen längst upp till vänster (Australien i detta fall) 2. Håll in (Ctrl/Cmnd+Shift) 3. Tryck på ➡️ tangenten (Släpp inte (Ctrl/Cmnd+Shift) 4. Tryck på ⬇️  tangenten (Släpp inte (Ctrl/Cmnd+Shift) 5. Voila alla celler som vi ska välja är valda
    ✅
    Max = 1,9267 → 192,67% Min = -0,174 → ‒17.4%
    ‣
    Answer

    A) ‒17.4% and 192.7% ✅

    B) ‒17.4% and 60.88%

    C) ‒17.4% and 120.78%

    D) ‒60.88% and 17.4%.

    E) None of the above (A, B, C, D) is close to be correct.

    Q8 - Find the highest pairwise correlation coefficient of quarterly returns

    The highest (maximum) pairwise correlation coefficient of quarterly returns is between?

    1️⃣  Create a correlation chart for the monthly returns 📗

    Tryck med musen 🐁 högst upp i excel menyn → Data → Data analysis → Correlation

    image
    ⚠️
    Ett nytt blad i excel skapas → Bladet blir inte automatiskt färglagt 🎨  Det nya bladet innehåller siffrorna

    2️⃣ “Färglägg” bladet 🎨 

    Tryck med musen 🐁 Home → Conditional Formatting → Color Scales → Välj en skala

    Färgskalan gör att du enkelt kan se vilka värden som är högst och minst Du kan ta bort ettorna, men om du är färgblind är denna metod inge bra ☹️ (Kör max funktion och ctrl-f:a ) 
    ✅
    Max → Nere i högra hörnet → Croatien + Slovenien Min → Finns två som är extra röda, den till vänster är mindre → Switzerland + Nederländerna
    ‣
    Answer

    A) Croatia and Slovenia.✅

    B) UK and US.

    C) Sweden and Norway.

    D) Israel and Slovenia.

    E) None of the above (A, B, C, D) is close to be correct.

    Question 9 & 10

    Publicly listed Real Estate Investment Trusts (REITs) have been a great investment for many years. The monthly historical REITs data is downloaded from the Monthly Index Values & Returns. We now focus on the index for All Equity Reits, Total Return and Index columns. See the yellow areas in column W and X.
    image
    image

    Q9 - Compute hypothetical Returns on monthly historical REITs data

    Suppose that you invested $10 000 in January 1998. What was the value of that investment in September 2021?

    1️⃣  Add column next to Return column

    2️⃣  Find Row that represents returns from Jan 1998

    → Row 323 in my excelfile

    3️⃣  Input 10 000 in that cell

    4️⃣  Feb 1998 is computed in picture → Then drag until September 2021

    ✅
    September 2021 → $82 022.
    ‣
    All answers

    A) $24 641. B) $82 022. ✅ C) $104 224. D) $720 000. E) None of the above (A, B, C, D) is close to be correct.

    image

    Q10 - Calculate Geometric Mean on Returns

    For the entire time period, December 1971 to September 2021, what is the monthly geometric mean return?
    🧠
    För använda formeln “=Geomean” behöver alla värden vara positiva - Så hur löser man det?

    1️⃣  Lägg till en Column för Geo mean

    image

    2️⃣  Första värdet börjar på 100 + r → Dra ned formeln hela vägen ner till september 2021

    3️⃣  Använda Geomean formeln

    =GEOMEAN(V11:V607)-100

    OBS 🚧  Svaret är i % - Vi tar bort 100 för vi la till 100 i steg 1)

    ✅
    Geomean = 0,93 → 0,93%
    ‣
    Answer

    A) 1.05%. B) 0.93%.✅ C) 12.05%. D) 2.25%. E) None of the above (A, B, C, D) is close to be correct.

    Questions 11-13 - Chi-Square stuff

    You want to study the market for exclusive loudspeakers. You have divided speakers into two categories: exclusive (above $3 000 for a pair of loudspeakers) and not exclusive (up to $3 000). You are interested to learn if there is a difference in the proportion of people who like the Dune (2021) movie and do not like the Dune (2021) movie who buy exclusive speakers. Based on a sample of people you have asked, you construct the following table:
    Actual Data
    Like Dune
    Don’t like Dune
    Total
    Exclusive Speakers
    108
    95
    203
    Not exclusive Speakers
    33
    88
    121
    Total
    141
    183
    324

    Q11 - Calculate upper-tail critical Value from a Chi-Square Distribution

    The upper-tail critical value from a chi-square distribution at the 0.01 level of significance is______.
    🧠
    To calculate a critical value of a chi-square distribution of OUR CURRENT data table we can use =chisq.inv.rt(alpha, df) in excel 📗
    ⚠️
    This is OK Because of: critical value = inverse = inv upper-tail = right tail = rt alpha = significance level = 0,01 = 1% = 1 df = degrees of freedom = (columns - 1)*(rows -1) = 1
    ✅
    =chisq.inv.rt(1, 1) = 6,634 upper tail critical value
    ‣
    Answers

    A) 6.63. ✅ B) 5.99. C) 20.74. D) 2.71. E) None of the above (A, B, C, D) is close to be correct.

    Q12 - Compute chi-square test statistic based on the collected Data

    Referring to the Dune table, value of the chi-square test statistic is ______.
    Actual Data
    Like Dune
    Don’t like Dune
    Total
    Exclusive Speakers
    108
    95
    203
    Not exclusive Speakers
    33
    88
    121
    Total
    141
    183
    324
    ‣
    1️⃣ We can turn the original data into the chart below 👇

    Example of how we calculate “Like Dune” Column

    108/144 = 0,77 33/144 = 0,23 144/144 = 1

    Actual Data Fractions
    Like Dune
    Don’t like Dune
    Total
    Exclusive Speakers
    0,77
    0,52
    0,63
    Not exclusive Speakers
    0,23
    0,48
    0,37
    Total
    1
    1
    1
    Predicted
    Like Dune
    Don’t like Dune
    Total
    Exclusive Speakers
    88
    115
    203
    Not exclusive Speakers
    53
    68
    121
    Total
    141
    183
    324

    2️⃣ From these tables we compute Chi-Square test statistic

    Exclusive Speakers Who like Dune: ((108-88)^2)/88 = 4,37

    Not exclusive Speakers Who like Dune: ((33-53)^2)53 = 7,33

    Exclusive Speakers + Don’t like Dune : (95-115)^2/115 = 3,37

    Not exclusive Speakers + Don’t like Dune = 5,65

    Summan av dessa är Chi-Square test statistic

    ✅
    Chi-Square test statistic: 3,37 + 5,65 + 7,33 + 4,37 ~ 20.74
    ‣
    Answers

    A) 6.63. B) 5.99. C) 20.74. ✅ D) 2.71. E) None of the above (A, B, C, D) is close to be correct.

    Q13 - Calculate P-Value of Chi-square test, Given chi test statistic

    The conclusion for a Chi-square test would be that ______.
    🧠
    Från fråga 11 vet vi att 6,6 är “Critical Value” → Alla P-Värden större än 6,6 P-Value =chisq.Dist.RT(chi-t-statistic, df) Upper-tail = RT Df = Degrees of freedom = 1 P-Value =chisq.Dist.RT(20,74, 1) → 5,27012E-06 (Mycket mindre än 6,6)
    ‣
    Answers

    A) The p-value must be higher than 10 %, implying that the null hypothesis will be rejected. B) The p-value must be higher than 10 %, implying that the null hypothesis will not be rejected. C) The p-value must be lower than 1 %, implying that the null hypothesis will not be rejected. : D) The p-value must be lower than 1 %, implying that the null hypothesis will be rejected. ✅

    Question 14 & 15

    Model Selection Criteria

    Model
    AIC
    SC
    RMSE
    1
    - 3,55
    -3,45
    0,85
    2
    - 3,66
    -3,49
    0,82
    3
    - 3,15
    -3,07
    0,92
    4
    - 4,42
    -4,28
    0,97
    5
    - 4,08
    -4,03
    1,05
    Suppose you have estimated five different (cross-sectional) econometric house price models and want to choose the one model you want to offer large commercial banks as the best model ever for predicting house prices based on their characteristics (so called hedonic house price models). ← The table summarizes the results for AIC, BIC and RMSE

    Q14 - Which model should you choose based on the AIC criteria?

    🧠
    SC and AIC models should be as low as possible, negative is good. RMSE can’t become Negative, hence close to zero is the best.
    ✅
    Model 4 → AIC = -4,42
    ‣
    Answer

    A) Model 1.

    B) Model 2.

    C) Model 3.

    D) Model 4. ✅

    E) None of the above (A, B, C, D) is correct

    Q15 - Which model should you choose based on the RMSE criteria?

    🧠
    SC and AIC models should be as low as possible, negative is good. RMSE can’t become Negative, hence close to zero is the best.
    ✅
    Model 2 → RMSE = 0,82
    ‣
    Answers

    A) Model 1. B) Model 2. ✅ C) Model 3. D) Model 4. E) None of the above (A, B, C, D) is correct.

    Question 16 - Exogeneity vs Endogeneity, how does these affect covariance

    In econometric modelling the assumption of strict exogeneity implies that_____.

    Picture of Covariance with strict exogeneity in the middle.

    image
    🧠
    Endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term. Exogeneity is the opposite of endogeneity, which an explanatory term is not correlated with the error term. This means the Covariance would be 0 if it’s strict exogeneity.

    ← Picture of Covariance with “strict exogeneity” in the middle.

    Answers

    A) cov(xi, ej) = 0: there is no correlation between the omitted factors (variables) associated with the observation j and the value of the explanatory variable for observation i. ✅ B) cov(xi, ej) ≠ 0: there is positive or negative correlation between the omitted factors (variables) associated with the observation j and the value of the explanatory variable for observation i. C) cov(xi, ej) > 0: there is strict positive correlation between the omitted factors (variables) associated with the observation j and the value of the explanatory variable for observation i D) cov(xi, ej) < 0: there is strict negative correlation between the omitted factors (variables) associated with the observation j and the value of the explanatory variable for observation i E) None of the above (A, B, C, D) is correct.

    Question 17, 19, 20 & 21 - Understanding regression

    image
    image

    Q17 - Compute R-Square for the model

    The R square for this model is closest to _________.
    🧠
    We can compute R-Square with “Multiple R” → (Multiple R)^2 = R-Square
    ✅
    0,7986^2 → 0.6377 = R-Square
    ‣
    Answer

    A) 0.362 B) 0.638 ✅ C) 0.568 D) Not possible to compute given the information in the summary output.

    Q18 - Compute the Standard Error for the regression model

    The standard error of the regression is closest to _________.
    Ulf Nilssons Magisk skiss
    Ulf Nilssons Magisk skiss

    1️⃣  Compute the Residual Error

    Residual: 642-409 = 232

    Residual Error = 232

    2️⃣  Compute SSE/df

    df = observations - Regression - 1 df = 2160 - 6

    SSE: 232/2154 = 0,107

    3️⃣ Compute the standard Error

    Standard Error: Sqrt(SSE/df) Sqrt(0,107) = 0,32

    ✅
    Sqrt(0,107) = 0,32
    ‣
    Answer

    A) 0.329 ✅ B) 0.799 C) 0.126 D) Not possible to compute given the information in the summary output.

    Q19 - How to compute t-statistic for linear model

    The t-statistic for the ln(SQFT) (see ln_sqft) is closest to_________.
    🧠
    t-statistic = (”ln_sqft_Coefficient”)/ (“ln_sqft_Standard_error”)
    ✅
    t-stat: 0,892479/0,015659 = 56,99
    ‣
    Answer

    A) 5.69 B) 0.00 C) 56.99 ✅ D) Not possible to compute given the information in the summary output.

    Q20 - Confidence interval for a variable in a linear model

    The 95% confidence interval for β5 is closest to__________.
    🧠
    We have the t-stat for b5 (waterfront) - With the t-stat and standard error we can compute the coefficient.
    t-stat
    Standard error
    Coefficient
    Two Tailed Critical t
    Interval
    Given
    6,682043
    0,020207
    6,68*0,02 👇
    =T.INV.2T(5%;2154) 👇
    =(Stand.err.)*(two-t Criticalt) 👇
    Calculated
    0,135024043
    1,961065926
    0,02*1,96 = 0,0396
    🧠
    95% confidence interval is closest to Lower = (Coefficient) - (Interval) → 0,1350 + 0,0396 = 0,1746 Upper = (Coefficient) + (Interval) → 0,1350 + 0,0396 = 0,0953
    ‣
    Answer

    A) (‒1.8260, 2.0961) B) (0.1148, 0.1552) C) (0.0954, 0.1747) ✅ D) Not possible to compute given the information in the summary output.

    Q21 - How to interpret a linear model

    Which of the following interpretations of the coefficient of WATERFRONT is most correct? A) A waterfront house in on average 1.35 times more expensive than not waterfront houses. B) A waterfront house in on average 13.5 % more expensive than not waterfront houses. C) A waterfront house in on average 13.5 % more expensive than traditional houses. D) A waterfront house in on average 135 % more expensive than not waterfront houses. E) None of the above (A, B, C, D) is close to be correct.
    🧠
    We calculated the Waterfront Coefficient to 0,1350 → Waterfront is 1 or 0, there fore a waterfront house is 1*0,135 more expensive than a none waterfront house. Assuming the compared houses for other attributes are identical we can derive that it will be 13,5% more expensive.
    ✅
    B) A waterfront house in on average 13.5 % more expensive than not waterfront houses.
    ‣
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