Can you downgrade to iOS 6.1.4 on an Iphone 5?

Running iOS 7.0.3

A step-by-step guide to help users restore their devices from iOS 7 GM to stable versions of iOS 6 (iOS 6.1.4 or iOS 6.1.3) via iTunes Backup and Restore function.
Pre-Requisites is Download iOS 6.1.4 for iPhone 5
(NOTE: It is not possible to downgrade the firmware from iOS 7 GM or final release to iOS 6.1.2 on iPhone 5, iPhone 4S and iPod touch 5. However, iPhone 4 users can still downgrade to iOS 6.1.2 if they have the required SHSH blobs and APTickets saved for 6.1.2 or lower firmware version.)

Step 1: Connect the iPhone, iPad or iPod Touch running iOS 7 GM or final release to your computer (PC or Mac) via USB cable. Then launch iTunes and place the phone into DFU mode (Hold the Power and Home buttons for 10 seconds, then release the Power, but continue holding Home button for 10 to 30 more seconds to boot the device into DFU mode).

Step 2: Choose your device from the left pane or click the iPhone button on the top right hand corner of iTunes.

Step 3: In the main iTunes screen, two buttons namely: Check for Update and Restore iPhone are visible. If running Windows OS then hold the Shift key (Alt Key for Mac) and click Restore iPhone.

Step 4: A new pop-up window should appear. Browse and select the IPSW file downloaded earlier for your iPhone, iPad or iPod touch and click Restore.

Step 5: iTunes will validate the IPSW firmware file with Apple and then the restore process will begin. This process may take some time to complete.So, please be patient and do not interrupt the process. Once the process completes, your device will be running iOS 6.1.3 or 6.1.4.

For further assistance, check out the video-tutorial

No, sorry. I know you can downgrade with a software called Tinyumbrella, but that's only for the older devices or if you have saved your SHSH file.

Its actuallty not easy process But you can try

Step 1: First up, you'll need to be running iTunes 11.0.5 or below, so if you're on a newer version like iTunes 11.1.x, you will need to essentially uninstall it first. This is because starting from iTunes 11.1, Apple has killed the ability to restore to custom firmware files from iTunes. When uninstalling, make sure you remove all traces, including services like Apple Mobile Device, Apple Software Update and Bonjour.

Step 2: After a restart to wash away those traces, download iTunes 11.0.5 from OldApps, or any similar such site [direct link]. Install it, and if you encounter an error relating to "iTunes Music Library.xml", head over to your iTunes Music folder and delete said file.

Step 3: Grab the latest version of iFaith (here), and also download iOS 6.1.3 or download iOS 6.1.2 firmware for iPhone 4, depending on which firmware you have saved blobs for to downgrade to. You will also need to retrieve your SHSH blobs from wherever you have happened to save them locally. If you previously had them saved on Cydia, you can make use of "Show Available SHSH Caches on Server" option in iFaith to save them on your HDD now.

Step 4: Fire up iFaith, and click "Build "*signed* IPSW w / Blobs". This will begin the process of creating your modified firmware file to which you will be downgrading.
Step 5: Locate your SHSH, and then select the stock iOS firmware file that you downloaded in Step 3 above.iFaith will then work its magic and create a custom IPSW.

Step 6: Follow the on-screen instructions to put your device into pwned DFU mode using iFaith's built-in DFU Pwner:

Hold Power and Home buttons at the same time for 10 seconds
Release the Power button whilst continuing to hold Home until the screen is blank.
Congratulations, you have successfully entered DFU mode.
Step 7. Finally, Load up iTunes, select your iPhone, hold down the left SHIFT key and click on Restore iPhone at the same time, and browse to that custom firmware file created by iFaith. The restore will then proceed.
Once the restore is complete, your iPhone 4 will be back to iOS 6.x.

Sorry, no.
But if you jailbreak your iPhone 5, I think you can.